''"=''\ of e^^^''":'-"'.) , and of
v/p^e'".'+'=''', v/?oe~<"''+°''- The constants ni , n'l and 1^2 may themselves be
developed in powers of e, i/y, ccq u'q and Bp.
277. Let us proceed to the second generalization method, and let
us assume that we wish to study the equations in the vicinity of a true
periodic solution having the form
ar, = ar, = i, = ^1 =^, = o.
We shall set
F = £> r, I, = ix\ , 7, = £/', , Tt = «a:',, Xt = E^;.
from. which it follows that
F = f;-+-sF', H-....
The equations remain canonical equations , and we have
where $ is a homogeneous quadratic form in -^i, y*! , X2, ^2- The co-
efficients of $ and h are periodic functions of ys = y'3.
However, we shall remove the accents which have become useless,
and we shall simply write
F, = hx,-i-'P(xuy,, T,,^,).
Just as in No. 274 and 276, it may be shown that we may always assume
that h may be reduced to a constant.
Let us now consider the equations
112
lU ~ ' dt dys. ' ^' ^'^i
rfr, _ rf* £Xi— _^.
~dc ~ dyt' dt dxf
They are linear and have periodic coefficients. Their general solution
will have the form
Xi = Aie"'?!.]-*- A,e-«',.,-i-y,B;j.,-(-a:',ij).,.,-)-y,9j.,,
Xi = ar'i l0g^=2P,,
x^
x\y\=li^, logZl==2C,,
3-,
and the calculation may be performed as was done in Nos. 275 and 276. /m
The following conclusion will be reached.
The x^'s and the y^'s may be developed in powers of e of three
constants oq, Bq s"*^ &'o» of e*'i'"i'+fj.i^ and of
The exponents n^ , n2 and n'2 may themselves be developed in powers of
e, ao. Bo and ?o-
This generalization may be directly applied when there are n degrees
of freedom. The first case, which is the case given in the preceding
section, corresponds to that in which there are n + 1 invariant relation-
ships and one single linear relationship between the mean motions. This
is what we discussed in Chapter XIX.
The second case, which is what we are discussing in the present
section, corresponds to that in which there are 2n - 1 invariant rela-
tionships describing a true periodic solution, and where there are n - 1
linear relationships between the mean motions. This is the case of asjraip-
totic solutions which we discussed in Chapter VII.
However, there are intermediate cases in which we have n - q invar-
iant relationships and q linear relationships between the mean motions.
Then the x-^'s and the y^'s may be developed in positive or negative
powers of q real exponentials and of n - q imaginary exponentials.
114
Relationship with Integral Invariants
278. Let us assume that the canonical equations
^.,^^F ,., ^^ (^.:, a, ...,«) (1)
dl dfi' dl dXi
have a periodic solution with the following form
where h is an integration constant. Let T be the period, in such a way
that (f>. and if), may be developed in series of sines and cosines of the
multiples of 2tt (t + h) .
T
Let us consider the solutions which are near this periodic j^ili
solution. According to the preceding statements, they may be written in
the following form: x^ and y^ will be developed in powers of 2n - 2
quantities which are conjugate by pairs, and which I shall call
A,e''.', A', e-^.' ■
Aje'.', A'.e-^.'
A„_,e»-.', A'„_, £-«-.'.
The A's and the A' 's are arbitrary integration constants. The ex-
ponents a may themselves be developed in powers of A^Ai, A2A 2. •••>
An-lVl-
In addition, the expansion coefficients of x^ and of y^ are periodic
functions of t + h, having period T. These coefficients (just like the
exponents a) depend, in addition, on the energy constant C.
We know that there is an Integral invariant
Jy:dT,dy,; (2)
from which it follows that, if g and y are two integration constants,
we must have
Zj\d? d-( d'l d<^ )
We could write this equation in another form. Let us assume that 3
is increased by 66, and that as a result for x-^, y^, A^e'*!
have the following increases:
115
o.r,-, oj',-, oA/e'.'
On the other hand, let us assume that y is increased by 6'y, and that
.ng incr
as a result we have the following increases for x , y..
• • t
Our equation may be written
- ( '-ri '.'vi — ?_/, f/.Ci) ^. consl. (3)
The second number is a constant. By this I mean that it is a
function of the integration constants multiplied by <56i5'y.
We obviously have /113
oAe« = e^'(3A + tZi).
On the other hand, we have
S^, = __ oC + -^^^- ok +2 ^Z(A, .-^) '^^ '^' -^li ^A:,..--!'-) -^* -^ ="''
M = -.y, oC -I- 7,7,-; -r^v o(AkAk)
(/(j .<- ia( Ak Ak)
rf(AKAii)
It can thus be seen that 6x and 6y have the following form
^yt = r^i + iM.i\ ^'yt -' ■>•,',• 4- t r/, f
3.r,- = 5,- 4- ; ^, ,. ; 3'^,. =^ ^'. 4_ ; J- ^^
where £.,£, ^.n.-n . are linear with respect to 6C, 5h, and to the
1 l.i 1 1.1
6Ae s and {A'e"'* s. In addition, they may be developed in powers of
the Ae*^^ s and the A'e'Ct's and the sines and the cosines of the multiples
of — (t + h) . The expressions of 6'x , i5'y. may be readily obtained.
It is sufficient to change 6 into 6 ' in those of 6x and (5y . . It may
be then seen that equation (3) may be written in the following form
D + E« + F^! = const.,
from which it follows that
F- S($,./l',..— ?'...■ ^, I..)
are developed in powers of the Ae""^^ A'e"*^^ s and the sines and cosines
of the multiples of — (t + h) , and they are bilinear with respect to
the
116
5'Atf-^', 3'A'e «', S'G, o7).
The first term must be independent of t, and we shall have
which has already provided us with certain verification relationships
which must be satisfied by the expansions of the Xj^'s and the y-^'s.
Thus, D must be independent of t. It will therefore be linear /114
with respect to the following determinants
aX^.o A'j. — o'AiAAi,
A'a A;(oAj.o'Ay — oAy o'A/,),
Ai.(cAA3'G-o'A;,i5C), (4)
Ai(oAi o'/j -- o'Ax-o/i)
(or with respect to similar determinants determined from the former
by interchanging Aj^ with A'j^, or A^ with A' .).
The coefficients will be developed in powers of the Aj^Aj^'s, and
will depend in addition on C.
The time must disappear. The exponentials must therefore disappear,
which can only happen if each factor Ae'^t is multiplied by a factor
A'e""^ or 6A'e-«t, or 6'A'e-"t.
A new series of verification relationships may thus be deduced from
this.
279. Among the ay^ exponents, some are imaginary, and others are
real. Among the real exponents, some are positive, and others are nega-
tive. However, since I may arbitrarily choose an exponent which I may
call ot]^ from between two exponents which are equal and have opposite
sign, I shall not limit the conditions of generality by assuming that Oj^
is positive if it is real.
Let us now cancel the coefficients Aj^^ which correspond to an
imaginary exponent, or to a positive exponent.
We will then have the following, if aj^^ is real
A;[. =0, A'^>o
117
^
x
and if a is imaginary
A^-^Al.
In addition, I shall set
where C is the value of the energy constant which corresponds to the
periodic solution under consideration.
Our series will then be convergent, and will represent the asymp-
totic solutions which we studied in Chapter VII. They include h and
the A^'s, which correspond to negative exponents, as arbitrary constants.
We shall therefore have 2n equalities which will express the x 's /llS
and the y. 's as functions of t and of these constants h and AJ. If we
eliminate t, h and the V's between these 2n equalities, we shall have
a certain number of invariant relationships between the y^'s.
If a group of values of the x 's and the y. 's is regarded as repre-
senting a point in space having 2n dimensions, these invariant relation-
ships will represent a certain subset V of this space. This is what I
shall designate as the asymptotic subset .
Let us reconsider the integral invariant
X d.Ti d/i
f^
and let us extend the integration over a portion of this asjnnptotic
subset V. In other words, let us assume that every system of values of
the x.'s and the y.'s, which form a part of the integration region,
satisfies our invariant relationships.
I may state that the integral invariant will be zero .
It is sufficient for me to demonstrate the fact that
and this is apparent , because we have
A/c — o, G = Co,
from which it follows that
N
118
SA^ = o, 3C = o,
S'Ai = 0, o'C = o,
which shows that all of the expressions (A) are cancelled. We can also
set
C — Co,
Ai^), A'<. = o (for real a^) ,
A4.= Ai = o (for imaginary a]^) .
We shall have obtained a new series of asymptotic solutions and, conse-
quently, a new asymptotic subset to which the same conclusions will
apply.
The procedure which we followed for the invariant (2) could be
followed for an arbitrary bilinear invariant (invariant of the third
type. No. 260), i.e., having the form /116
CCzudxidx^, (5)
where B is a function of the x^'s and of the y^^'s and where one or two
of the differentials dxj^, dx^ may be replaced by iy-^ or dyj^ under the
sign 1.
The expression
will still be linear with respect to the quantities (4) . This would
still apply to a quadratic invariant (invariant of the second type.
No. 260) having the form
J^/I^Td^^d^t, (6)
where B is a function of the x^^'s and the y^j^'s, and where one or two of
the differentials dx^, dxj^ may be replaced by dyj^, dyj^ under the sign Z,
It may be seen that the expression
SB IxiZxk
must be linear with respect to the expressions
/ SA^oAi,
) Ai.A}5Ai8Ay, ^^1 J
Ai8AiSC,
SCSA
119
\
and to those which may be deduced from them when interchanging Ajj^ and
A'l^., Aj and A' j .
For every asymptotic subset, the invariant (5), like the invariant
(6), must be cancelled.
Another Discussion Method
280. This same study may be pursued farther, while presenting it
in a different form.
For example, we shall assume that we are dealing with a problem of
dynamics, that the x^ s are the coordinates of different points of
matter of the system, and that the conjugate variables y^ are the com- /117
ponents of their momentum. We plan to study the integral invariants
which are algebraic with respect to the Xj^'s and to the y^^'s, and to
determine whether one may exist in addition to the one which is known,
and which is written
//
S dxi dyi.
We have seen that, in the vicinity of a periodic solution, the x^'s
and the y^'s may be developed in powers of the Ae^^'s, . . . . We are going to
consider these expansions again, but we shall assume that the value of
the energy constant corresponding to the periodic solution is zero,
so that the expansions will not only proceed in powers of the Ae"*^'8,
but even in powers of C. In addition, they will depend on t + h.
By equating the Xj^'s and the y^'s to these expansions, we obtain
2n equations, which we shall solve with respect to the Ae^'^'s, C and
t + h.
We have
G = '!>,
We should point out that oq . like a.^, may be developed in powers
of C and of the A^/iij^'s. It may be seen that fj^^, f^, $, cosG, sine are
uniform functions of the x^'s and the yj_'s in the vicinity of the
periodic solution. In addition, the Xj^'s and the y^^'s may be developed
120
V-^
in powers of the fj^^'s, the f^'s, and $, and according to the sines and
cosines of the multiples of 0.
On the other hand, the expression
which corresponds to the invariant (2) , or the similar expressions which
would correspond to another bilinear invariant of the form (5) , must be
developed in powers of the f^^, fj^, $'s and be bilinear with respect to /118
Vk, s/i., S'J>, Se,
«'/*. o7i-> S', o'e.
In addition, when we replace fj^, f'j^, $, by their values (7),
this expression must be independent of t. The time t may be introduced
in three different ways:
1. In the exponential form;
2. In the form of the cosine or sine of the multiples of (t + h) ;
3. Outside of the exponential and trigonometric expressions (and,
as we shall see, of the second degree and more).
It must not enter in any of these three ways.
1. In order that it does not enter in the exponential form, it
is necessary and sufficient that the expression be linear with respect to
the following quantities which are similar to (4)
^/k ^''f'k — ^'fk ?/I,
/U3Ao''l>-o'/^.5'^), (8)
where the coefficients may be developed in powers of the f^^^'s, fj^^'s, and
of $.
2. In order that t does not enter in the trigonometric form,
it is necessary and sufficient that our expression does not depend on
0, but only on its variations 60, 6'0.
121
3. We must now determine the condition under which t does not enter
outside the exponential and trigonometric expressions. We should point
out that we have
We may distinguish five types of terms in our expression, depending
on whether they contain as a factor a quantity (8) included in the first,
second, third, fourth, or fifth line of the table (8).
Under this assumption, if we replace 6f , ... by their values (9),
we shall see that the five types of terms include as a factor, respec- /119
tively,
(3A*8'Ai— SA'*S'A*)H-<[8»«S'(A*.\'*)-S'a*o(A*Ai)],
A'*A}(8A*o'Ay-8AyS'At)
+ AiA)<[A*(Sa*8'Ay— o'a*5Ay) — A/(Sa/S'A*-o'aySA*)]
-1- AxrAiAyA;t«(Sat8'a; — SayS'.ai),
( A'*(SA*8'C — S'A*8C)H-A*AVtCS«*8'C-8'a*SC), ^q^
A'i(8A*S'Po-S'A*8fJo)-i-A*Ai«(8iA-8'Po-8'a*8Po)
+ Ai/(8A*3'oo — S'AA■8ao)-^- AiAi^Voiio'"*— o'"*^"*).
(SCS'P,— 8'C8?o)+<(8GS'o,-S'C8I(,).•
It may be seen that the time can enter in the second power.
Let us first make the terms for t^ disappear. They may only begin
with terms of the second type or of the fourth type.
It may be stated that the coefficient of
i'(Sat8'ay — 8ay8'at)
must be zero.
In actuality, due to the fact that the virtual displacements in the
constants are arbitrary, we may asstraie that all the 6a. 's vanish, with
the exception of 6a , and in the same way it may be assumed that all the
iC
6'a. 's vanish with the exception of 6'a .
All the terms in t^ cancel, with the exception of the term in
«»(8a*8'ay — 8a/8'i^).
There would be an exception if there were a relationship between the
n - ] exponents o . . We could no longer assume that all the 6a 's
122
cancel except one, unless the last one itself cancels too.
There are now four terms of the second type which result In terms
For purposes of brevity, I may write them in the following form
l};! (Ill -H i)*! "> "*" 't' J "» *+■ 4'k "* >
The ijj'sare developed in powers of the f^, f'^ and of $. I have employed
iMl to designate the expression which appears in the second line of the
table (8): 11^
0)2 may be deduced from ui by interchanging f^^- and f'^,
0)3 may be deduced from o)i by interchanging fj and f'j,
0)4 may be deduced from 0)^ by making these two permutations at the
same time.
In order that the terms in t^ disappear, it is necessary and suffi-
cient that
'I'l — 'J's — "^i -+- ^i =0.
If this condition is fulfilled, our four terms
will provide us with the following terms in t
(11)
-i-{^, — 6t)t \j A;[ Ssy o'( Ai- A'i) — o' ay o( A/, A'^.)].
Let us now consider terms of the fourth type, which we shall group
together by pairs. Let the following be one group of two terms
where fi and i|j2 may be developed in powers of C and of the Aj^^Al^^' s, where o)^
is the expression included on the fourth line of the table (10) , and where
0)2 is that which is deduced by interchanging A^ and AlJ^ and changing a^^ to
In order that the terms in t^ disappear, it is necessary that
4'i = ).
becomes. We must also have identically
-l-SO*[o(AxAi.)&'au-o'(AiA'i.)oj„l-f-D„(oG'!'ao-o'Coao) = o.
For purposes of brevity, let us write Yj^ instead of \^^, yo instead
of C and
instead of
We have
or
d(u, v)
ouo'c — ZvZ u ;
SSBt ^ d(v^, Yt.-)+ S2Di ^° d(Y,, Yy)-4- SDo ^ ^(>, y> ) = o.
"7/ "Ty "".'y
Under the sign H or EE, k may take on the values 1, 2, ..., n-1 and
j may take on the values 0, 1, 2, ..., n-1.
When setting the coefficient of 3(y4, Yfc) equal to zero, we obtain
d-(j d-'^k a-zj d-^k
124
By setting the coefficient of 3(yo. Yj) equal to zero, we have
"'To ' d-i, " d-^j (12')
These equations indicate that
is an exact differential.
We must set yj = in equations (12) and (12'). The ^'s are there-
fore constants. The a.'s are therefore linear functions of the y's.
In actuality, as we have seen, the a's may be developed in powers of ^ /122
the y's. However, the result which we have just obtained is only valid
if we neglect the squares of the y's, and if we stop the expansions of
the a's at the terms of the first degree. In addition, the B's and D's
are constants. The expression (13) is therefore the exact differential
of a polynomial of the second degree.
In order to carry this investigation further, let us express the
of
7o, 7i, •■■> Ti-i.
a 's not only as functions of
k
but also as functions of
^'o, Yi. •■•, Y«-i'
In order to avoid any confusion, let us employ 3 to designate the deriva-
tives chosen with respect to the new variables, and the d's to designate
the derivatives chosen with respect to the old variables.
It may then be seen that
is an exact differential, which entails the following conditions
„ d^^ „ rJsf, (14)
'h - - = L>/ , -•
If one knew the relationships between the a's and the y's,
these equations would allow us to determine the coefficients B^^.
We can express ^D^Y-j as a function of the variables
'oi Ti' Ti' •••> T"-i
while writing
125
\
The E, 's will be given by the equations
and Eq may be chosen arbitrarily.
It is necessary that equations (14) be compatible, which requires
certain conditions in the case of n > 3
doi dxi doj _ Ja£ dij d^_ (15)
'o^i O'lj o^k " ''r* ''y' '^ij
These conditions (15) will always be fulfilled, since there is
always an integral invariant /123
/=
tlxidyj.
If there are several integral invariants which do not vanish iden-
tically for the periodic solution under consideration, a system of
values of the coefficients Bj and E. must correspond to each of these
invariants .
If equations (14) have q solutions which are linearly independent,
we may calculate the corresponding values of the E^'s by means of equations
(14'). Since Eq remains arbitrary, we shall have q + 1 systems of values,
which are linearly independent, of the coefficients B. and E .
We may therefore have q + 1 different integral invariants (if the
periodic solution under consideration is not singular, with the meaning
attributed to this word in No. 257), but we cannot have any more.
282. I stated above that conditions (15) were definitely fulfilled;
there may still be some doubt on this point. If equations (14) have q
different solutions, we may have q + 1 invariants. If there is only one
invariant, we could assijme that q = 0. The presence of a single invariant
/
'S.dxidyi
would not enable us to state that equations (14) definitely have a solu-
tion.
This is the doubt which I wish to dispel.
I would first like to note that in the case of the three-body
126
problem, there are not one, but two integral invariants.
In Volume I, Chapter IV, we studied the variational equations of
this problem.
On pages 170 and 172 we obtained the following integrals
yui —y
^ m Jmi dx
In the same way, we could obtain
d^ , , (1)
5 = consl.
(2)
7124
^yr/ yr^dV „ . (1')
> <—^ — >—- f = const.
^ III ^.i dx
Let us multiply (2') by (1), d') by (2), and let us subtract. We
then have
^d\m dx I ^g^
^^/Zl _ g t') 1(^3.^, _t-75) = consl.
The first term is linear with respect to the determinants having the
form
We therefore have an integral of the variational equations, and we
may deduce from it a new bilinear integral invariant.
In the case of the three-body problem, we therefore have at least q=l,
and it may be stated that conditions (15) are fulfilled.
283. Is this still true in the general case? Let us assume that it
is not. Then all the coefficients which we have called B^ must be zero,
as well as all of the Ej^'s, with the exception of Eq.
Therefore, when we attribute the values corresponding to the periodic
solution under consideration to the Xi's and the yi's, i.e., when we set
C =A,--u e,c>,)
However, this is impossible, since the first term is a bilinear form
with determinant 1, and the second is a bilinear form with determinant 0.
We must therefore conclude that conditions (15) are always fulfilled.
284. Let us now try to determine whether equations (14) may have
several solutions.
Let
I^'.- BV B'
be these two solutions and let us assume that we do not have
B,-
and then the two equations
Hi — — = Ui -— ,
128
will imply
Then the indices
I, ..., 2, n
will be divided into a certain number of groups, as many groups as there
Bi
different values for the ratio rr". Two indices will belong to the same /126
group, if they correspond to the same value of the ratio -rf-.
^i
In order that o.^ depends on Yi (or a^ on Y]^) > it is necessary that
the indices i and k belong to the same group .
In order to formulate these ideas clearly, let us assume that there
are only two groups containing the indices, respectively,
Then
will depend only on
and
will depend only on
I. ' P,
/) -r- I , /) 4- 2, . . . , n — I .
'0. vi. Te. ••■, r/'.
'o, T/..-1. V*-"-' ■•■. T"-!-
It then appears that the characteristic exponents aj^ form several indepen-
dent groups, in such a way that the aj^^'s of one group do not depend on the
products AjA'. corresponding to another group.
The periodic solutions for which this condition will be produced
(or for which there would be one relationship between the a^'s) may be
called particular solutions.
We therefore arrive at the following conclusion:
In order that there be another algebraic invariant, in addition to
129
%
V
those which we know, it would be necessary tha t all the periodic solu-
tions be particular solutions, or thau they all b e singular solutions.
with the meaning KJven in No. 257 .
I shall not try to demonstrate the fact that this condition could
not occur in the three-body problem, but this would seem to be very
unlikely.
Quadratic Invariants /1^7
285. Let us now study the quadratic invariants from the same point
of view, i.e., the integral invariants having the form
//p.
where F is a quadratic form with respect to the differentials dx^, dy^.
Let us set
where the H's are functions of the x's and the y's, and where the product
dx dx may be replaced in certain terms by the product dx^dy^^ or dy^dy^^.
X K
We may then write the following equation which is similar to equation
(3) of No. 278
ZMoxi^ivj; — const. H)
On the other hand, we find in No. 278 that
oxi ^ ^, + l^,h 3/,- =--= Tj, + tr,,j.
We may then write equation (1) in the form
D ■(- lie + F r- -= con?t.,
where D, E, F may be developed in powers of the A^"*^ s , A^"""^ s and of the
sines and cosines of the multiples of y^ (t + h) , and where D, E, F are
quadratic with respect to the
oXc-J-', o.S.'e-=", ZC, ?,h.
We must therefore have
E .--: V r. O,
and, in addition, D must be independent of t, which shows that D must be /128
130
linear with respect to the following expressions
A';. oAa- oC,
oG o/i,
or with respect to the expressions which may be deduced by interchanging
k]^ and A!y., or Aj and A'^ .
The coefficients will be developed in powers of the products k■^^\
and of C (if one assumes that the periodic solution corresponds to the
zero value of the energy constant) .
286. Let us return to equations (7) given in No. 280, and let us
pursue the same line of reasoning as given in No. 280. We shall find that
the expression
n .-^ I II or, or,,
must satisfy the follow \ig conditions when the x^'s and yi's are replaced
by their expansions as functions of the fj^, fj^^, $ and s;
1. It must be linear with respect to the following quantities:
(8')
fi/'j 5/a- 5/;.
o'l. 00,
o'l.v^e'
where the coefficients are developed in powers of the f^ffe's and of $.
2. It will not depend on 0, but only on 60.
3. If these conditions are fulfilled, expression n will not include
the time, neither in the exponential form nor in the trigonometric form.
We must now determine the condition under which the time is not
included outside either the exponential or trigonometric terms.
Let us consider equations (9) again from Section No. 280. We shall
find that the following terms will correspond to the different terms UAl
given in the Table (8'):
131
N
A'^.A^oAi o\j-hA'/,:Vjt(\),Zn. SA; •+- XjOxj oA*)
A^. oAioC h X^.X'^to:ij,?jC, (10')
A'i.oA^o3o + A:iV(oA^.oao + AiOaiO,3o)-f-Ai.A'^.;'5i^8i(,,
Let us first make the terms in t^ vanish.
The entire group of these terms is a quadratic form with respect to
This quadratic form must be zero.
The coefficient of Saj^Sa^t^ must therefore be zero. However, there
are four terms which could introduce the product t^6aj^6a^ ; these are the
terms in
fkfj^^j^M,, /*/.?/* 5/;, fkfjZfWj, hfjinyy
For purposes of brevity, let us designate these four expressions by
loi, U2, "3, ui+. The entire group of our four terms may then be written
where i^^, i|/2 , ^-t, and ^b, may be developed in powers of the fj^f'j<.'s and of $.
In order that the coefficient of t^Saj^^Saj vanish, we must have identically
'l'l+ 'J-S + '5-3 + ■>» = O.
In the same way, the coefficient of t^S^aj^ must vanish.
It arises from terms in
Va-/i-, si^n, nyi-
For purposes of brevity, let us designate these three expressions by
'*''l » ^li ^Zy s'^d the entire group of the three terms by
where i|j\ , 1J/2, ip'a may be developed in powers of the fj^f, 's and of $.
In order that the coefficient of t^S^Uj^ may vanish, we must have /130
(11)
132
For the periodic solution, we must have
fi=f\ =/l =/*« = •■• =fn-\ =/'/.-( - o.
All the terms including as a factor one of the expressions appearing
on the 2nd, 3rd, or 4th lines of the Table (8') must then vanish,
because each of these expressions includes fj^ or fj^ as a factor.
The only terms of expression IT which do not vanish for the
periodic solution are therefore the terms in
Equation (11) shows that ^i contains fj^fj^. as a factor. Therefore,
the term i|; j 6 f j^^S f j^ must also vanish. We then have only the terms in
The first does not Include t, the second includes it in the first
power , and the third includes it in the second power.
Due to the fact that this third term is the only one which includes
t^, it must be zero. If it is zero, the second term will also be zero,
due to the fact that it is the only one which includes t.
Finally, all the terms of IT vanish for the periodic solution,
except the term in 6i|>^.
In the general problem of dynamics, just as in the case of the three-
body problem which we have designated as the restricted problem, the general
reduced problem, and the planar reduced problem , we have a quadratic in-
variant, but no more than one.
I may write the energy equation in the following form
F :-= const .
This invariant is nothing else than
and the term in 6$^ which does not vanish corresponds to this invariant.
If there is a quadratic invariant, other than that which is known, /131
this invariant must vanish for all points of the periodic solution.
In other words, this periodic solution must be singular in the sense
of the meaning given in No. 257.
133
There would be an exception, if the n exponents
'^a, o,, or,, ..., a„_i
were not independent of each other, but if there were one relationship
between them. In this case, the coefficient of t^, which is a quad-
ratic form with respect to the n variables
oil), Sli, ..., Olfl-i,
could vanish without all of its coefficients being zero, since
these n variables will no longer be independent.
To sum up, in order that there may be other quadratic invariants, in
addition to those which we are acquainted with, it is necessary that all
periodic solutions be singular or particular .
It is very unlikely that this will be the case for the three-body
problem.
Case of the Restricted Problem
287. We may conceive of another discussion method which we shall only
apply to the case of the restricted problem. The discussion presented in
No. 257 has presented the possibility of two quadratic invariants, of which
one is known. Let us assume that these two quadratic invariants exist, and
let II be the quadratic form corresponding to one of these invariants.
According to the preceding statements , IT may include terms in
( y.5/,, /;2/.S'^> /.^/i^'i-. /,o/.«9, /.3/,59, (j^)
On the other hand, IT is a quadratic form with respect to the
quantities
o.f,, Zxi, Zyi, o/j,
whose coefficients are the algebraic functions of x^ , x^, y^ , y^- /132
Following are the variables x and y which we shall select. In this
problem, which I have called the restricted problem, two of the bodies de-
scribe concentric circumferences, and the third (whose mass is zero) moves
in the plane of these circvraif erences . I shall refer this third body to
moving axes turning uniformly around the center of gravity of the first
two. One of these axes will constantly coincide with the line joining
these two first bodies. I shall use x^ and x^ to designate the co-
ordinates of the third body with respect to these moving axes, and
y, and y to designate the projections of the absolute velocity on the
134
moving axes .
Let us then set
'!> r^ F -+- (0 G,
where F and G designate the energy function and the area function
in the absolute motion, and where o) designates the angular rotational
velocity of the two first bodies around their common center of gravity.
The equations take the canonical form
(/.r,- _ fM> dfi _ c/•^
dt ~ dfi ' dt "' dxj
The integral $ = const, is nothing else than "the Jacob! integral"
(see Volume I, No. 9, page 23).
Under this assumption, our expression n will be a quadratic form in
o.r,, ox,, y',, ly~,
for which the coefficients will be algebraic in x^ and y^. If we assume
that the four variables x and y are related by the relationship
•!' ..= const.,
which entails the following condition
o'^ r^ o,
our four variables Sx^ , 6y^ will no longer be independent. One of them
could be eliminated, and n will become a ternary quadratic form.
Let us consider one point of the periodic solution. For this point,
we shall have
All the expressions (1) will therefore vanish with the exception of /133
5/,?/;, Be', S'l"'*? and ''"'■
If we set 6$ = 0, they will all vanish with the exception of
Vi'^f\ and 5e'.
Therefore, for a point of the periodic solution, let us set
n--=B5/,J/', + Coe'.
The entire group of terms for t^ will therefore be reduced, for
135
this same point, to
(see, supra. Table 10') and, since fi = i\ = 0, may be reduced to
CrSaJ.
The terms in t^ must vanish. The latter is the only one which does
not vanish for the point under consideration; all the others are zero,
even when the condition 6$ = is not imp osed , because &'^6Q and 6$ do not
_
provide terms in t^.
However, Solq is not also zero. For one point of the periodic solu-
tion, we have
'J/\~"J/\'''d§~ °'
dao
but we cannot be sure of having = 0. This would assume that there
d$
is a continuous infinity of periodic solutions having the same period,
which does not occur.
dao
Nevertheless , it may be noted that -jT~ includes the small quantity
which I may designate by y as a factor, i.e., the mass of the second
body. Consequently, it may be noted that 6ao vanishes for y = 0,
i.e., in Keplerian motion.
The terms in t^ can only vanish if we have
from which it follows
11 =B3/,3A.
However, this latter equation would indicate that II may be reduced
to a binary quadratic form and, consequently, that its discriminant is
zero. Thus, the discriminant A of n must vanish for every point of /134
every periodic solution .
288. However, an algebraic relationship such as
cannot be valid, unless it is reduced to an identity, for every point of
every periodic solution.
If the relationship a = o
136
is supplemented by two other relationships
F-P, G-v (3)
(where 3 and y are two arbitrary constants, and F and G are the two
functions which were designated in the preceding section) and a fourth
arbitrary algebraic relationship
H-o, (4)
the number of solutions of these four algebraic equations will be limited
whatever the constants g and y may be.
Let us now consider a periodic solution, and the variables x^^ and
y^ will be developed in powers of y in the following form
( ,r/ — x'i -h iixl -h. . .
In the same way, F will be developed in powers of y, and we shall
have
and G and H will be independent of y.
The quantity A remains. It may be stated that this function, which
is algebraic in x^ and y^ under the terms of the hypothesis, also depends
algebraically on y.
If we state that
f^n
is an integral invariant, we will be led to certain relationships which
include the coefficinets of II , their derivatives, and the coefficients
of the differential equations of motion.
We assumed that n is an algebraic function of the Xj^'s and the y.'s.
We may assume that this algebraic function is included as a special case /135
in a definite type, not containing y explicitly, but depending algebraically
on a certain number of arbitrary parameters. The quantity / /jf will not
be an integral invariant no matter what these parameters may be, but only
when these parameters take on certain special values depending on y .
When stating that / v^ is an Integral invariant, one Is led to
certain algebraic equations between y and these parameters. These equa-
tions must be compatible, and it is apparent that the parameters will
be obtained as algebraic functions of y.
137
The coefficients of the form II and A will also be algebraic in y.
The equation A = is therefore algebraic in y, and we may assume
that it has undergone a transformation in such a way that the first term
is a whole polynomial in y.
We may therefore write
In addition, Aq will not be identically zero, unless A is. If Aq
would vanish, A would contain a factor y which could be made to vanish.
The function A must vanish when the x^'s and y^'s are replaced
by the expansions (5). It may then be developed in powers of y and, due
to the fact that the term which is independent of y must vanish we shall
have
(2')
We should now point out that we must have
I G(:r°,/°)--Yo. ^^ ^
where Bq ^^^ Yo ^^^^ constants. In order that this may be the case, it is
sufficient to recall that, for y = 0, the motion may be reduced to
Keplerian motion.
Now, for example, let us take /136_
and let us write the equation
(x^l' + Ca-?)'^ r. (4')
If we set y = 0, we may then observe that the third body will describe a
Keplerian ellipse. Let 5 and n be the coordinates of this body, not with
respect to the moving axes, but with respect to the axes of symmetry of
this ellipse.
The equations of the Keplerian ellipse will then be written
I ,) = T,isinijj + T,jsin2tp +
The coefficients E.^, r]^ will depend on two constants which are the
major axis and the eccentricity of the ellipse and, consequently, on Bq
138
and Yo • We shall have
o = 7i| ? 4- raj,
where the mean motion ni depends on Bq and where a)^ is a new integration
constant.
The intersection of the ellipse (6) with the circle
^5 _1- T,» = I
will occur at two points which will be given by the equations
$ = cosO, T, = +5inO, ip = ±(po.
We will then have
(7)
:f2 = ^ sin (to ; -i'W,)— r, cos(io? -1- ra,),
(8)
where (1)2 is a new integration constant.
We shall obtain solutions of the equation (4') by combining equa-
tions (7) and (8) , which yields
(k is an arbitrary whole number) .
In order that the solution be periodic, it is necessary and suf fi- /137
cient that the ratio -^ be commensurable. Let us write this ratio in the
form of a fraction reduced to its most simple expression, and let D be
its denominator. It may be seen that equation (4') has 2D different
solutions.
Equations (2'), (3'), and (4') must have only a limited number of
solutions, no matter what the constants Bq and yo ™ay be. I may choose
60 in such a way that -^ has the value which I desire, and consequently
that D may also be as large as I desire.
This can only occur if Aq, and consequently if A, are identically
zero.
Consequently, the discriminant having the form n is identically zero,
and this form must be reduced to a binary form.
139
It could be shown in the same way that, in the sense of No. 257,
it is impossible that every periodic solution be a singular solution.
This has only been proven in a very special case, but it is possi-
ble that this proof may be extended to the general case.
289. The form 11 regarded as a binary form, must be reducible to
for one point of a periodic solution. The binary form will therefore be
definite (i.e., equal to the sum of two squares) if the periodic solution
is stable — i.e., if the characteristic exponents are imaginary. It
will be indefinite (i.e., equal to the difference of two squares) if the
periodic solution is unstable — i.e., if the characteristic exponents
are real.
Let us assume that y is very small, and let us reconsider equation
(4') .
According to the principles outlined in Chapter III (No. 42), for a /138
given value of Bq, we shall have at least two periodic solutions, of which
one is stable and one is unstable. Let
be the corresponding values of the constants Hi and (^2.
Let us set
Oh (?o — ='i) + '^j^Y.
9-i- — (?o—ran + '^s = •!''.
and equation (4') will give us, for the first periodic solution,
:r; = COS I i)/ -\ 1
and for the second
Without restricting the conditions of generality, we may assume that
^" > i|j' and that ijj' and \p" are contained between zero and il. Then the
D
form n will be
1)40
definite for x]---. cosf^ -h"^)
indefinite for x\ r.-. cos U" i- ^^j
definite for t\=. cos^^y -^- -^-
indefinite for -!== cos(,/+ i^^:
definite for :r;== cos(i>' + 2:1),
indefinite for .rj -- coscl-' +2-);
which shows that the discriminant of H, considered as a binary form,
must vanish at least 2D times. Just as above, It may be concluded
from this that it is identically zero.
The form n may therefore be reduced to a square term. Therefore,
since it must equal
B 0/, o/'i
for every point of a periodic solution, it must vanish for all of /139
these points.
The same line of reasoning would show that it is identically zero.
To sum up, there is no other quadratic invariant except the one which
is known, at least for the special case of the restricted problem.
141
CHAPTER XXVI
POISSON STABILITY
Different Definitions of Stability
290. The word stability has been understood to have several dif- /140
ferent meanings, and the difference between these meanings is clearly
apparent if we recall the history of science.
Lagrange has shown that. If the squares of the masses are neglected,
the major axes of the orbits are invariant. This means that, with this
degree of approximation, the major axes may be developed in series whose
terms have the following form
where A, a and 6 are constants.
If these series are uniformly convergent, this results in the fact
that the major axes are contained between certain limits. The system of
stars cannot therefore pass through every situation which is compatible
with the Integrals of energy and area, and furthermore It will repass
an infinite number of times as close as desired to the initial situation.
This is complete stability.
Carrying the approximation further, Poisson demonstrated that the
stability continues to exist when one takes into accout the squares of the
masses and when the cubes are neglected.
However, this does not have the same meaning.
He meant that the major axes may be developed in series, containing
not only terms having the form
A sin(a^ -h jB),
but also terms having the form /141
\liin{at-i- P).
The value of the major axis then undergoes continuous oscillations,
but nothing indicates that the amplitude of these oscillations does not
increase indefinitely with time.
We may state that the system will always repass an infinite number
142
of times as close as desired to the initial situation. However, we may
not state that it does not recede from it very much.
The word stability does not therefore have the same meaning for
Lagrange as for Poisson.
It is advantageous to point out that the theorems of Lagrange and
Poisson include one important exception: They are no longer valid if
the ratio of the mean motion is commensurable.
The two scientists concluded from it that stability exists, because
the probability that they are commensurable is infinitely small .
It is therefore advantageous to provide an exact definition of sta-
bility.
In order that there be complete stability In the three-body problem,
the three folowing conditions are necessary:
1. None of the three bodies can recede Indefinitely;
2. Two of the bodies cannot collide with each other, and the dis-
tance of these two bodies cannot desend below a certain limit;
3. The system repasses an infinite number of times as desired to
the initial situation.
If the third condition alone is fulfilled, without knowing whether
the first two conditions are fulfilled, I would say that there is only
Poisson stability .
A case has been known to exist for a long time for which the first
condition is fulfilled. We shall see that the third condition is ful-
filled also. I can say nothing with respect to the second condition.
This is the case given in the problem of Section No. 9, where I
assumed that the three-bodies move in the same plane, that the mass of
the third is zero, and that the first two describe concentric circumfer-
ences around the common center of gravity. For purposes of brevity, I
shall call this the restricted problem .
Motion of a Liquid /142
291. In order to provide a better explanation of the principle un-
derlying the proof, I am now going to present a simple example.
143
\
Let us consider a liquid which is enclosed in a vessel having an in-
variable form and which is completely filled. Let x, y, z be the coor-
dinates of a liquid molecule, u, v, w the velocity components, in such a
way that the equations of motion may be written
1^ ^ ^ = f^ :. ,/,. (1)
The components u, v, w are functions, which I assume to be given
functions, of x, y, z and t.
I shall assume that the motion is steady, in such a way that u, v, w
depend only on x, y and z.
Since the liquid is incompressible, we shall have
t/u (!v d>v
. — t- - -,- - - -„.
ilx dy dz
In other words, the volume
is an integral invariant,
/
Lv dy dz
Let us study the trajectory of an arbitrary molecule. I may state
that this molecule will repass an infinite number of times as close as
desired to its initial position. More precisely, let U be an arbitrary vol-
ume inside of the vessel, which is as small as desired. It may be stated that
there will be molecules crossing this volume an infinite number of times.
Let U be an arbitrary volume inside of the vessel. The liquid
molecules which fill this volume at the time will fill a certain volume
U, at the time t, a certain volume U„, ..., at the time 2t , and a certain
volume U at the time nx.
n
The incompressibility of the liquid or, which is the same thing,
the existence of the integral invariant, indicates to us that all the
volumes /143
Uo, U„ Uj u,
are equal.
Let V be the total volume of the vessel, and if
V<(n-i-,)U<„
we shall have
144
V i) . Let M
be a point in common to U^ and U^.. The molecule which is at the
point M at the time ix is, at the time 0, at a point Mq belonging to Uq,
since the point M belongs to U^.
In the same way, the molecule which is at the point M at the time
kx is, at the time (k-i)T, at the point Mq, since the motion is steady.
On the other hand, it is at the time at a point Mi belonging to Uq,
since M belongs to U^, and we must conclude from this that Mq belongs to
Uk-i-
Therefore, U^ . and Un have points in common.
•^"J- q.e.d.
There fore, it is possible to choose the number a in such a way that
Uq and U^ have a part in common.
Let U'o be the part in common, and let us form If i , U2, ■••. with I^q,
as we formed Ui , U2 , . . • , with Uq . We may obtain a number g in such a
way that U'o and U'g have a part in common.
Let If' be this part in common.
We may obtain a number y in such a way that ifj and IT^ have a part
in common.
This procedure may then be continued.
As a result, U'q is part of Uq, U" of U'o, and U'q" of U',], ... In
general, uJP"^^^ will be part of U^P^ When the number p increases in-
definitely, the volume uj^^ must therefore become smaller and smaller.
According to a well-known theorem, there will be at least one Point,
perhaps several, or perhaps an infinity, which belong at the same time /144
to Uq, to U'o, to U'o' , ..., and to ujp) , however large p may be.
145
This group of points, which I shall call E, will be in a measure
the limit toward which the volume u(p) tends, when p increases indefin-
itely. ^
It may be composed of isolated points; however, it may be somewhat
different. For example, it may happen that E is a region in space
having a finite volume.
A molecule which will be inside of U^, and, consequently, inside
of Uq, at the time zero, will be inside of Uq at the time -ar.
A molecule which will be inside of U'J and, consequently, inside
of U' at the time zero, will be inside of l/g at the time -gx, and,
consequently inside of Uq at the time -(a + B)t.
A molecule which will be inside of U'J' at the time zero will be in-
side of U'(j at the time -yx, inside of Ujj at the time -(g + y)t, and in-
side of Uq at the time -(a + 6 + y)t.
Since Uq" , Uq , U'q are part of Uq , this molecule will be inside of
Uq at four different times (multiples of t) .
In the same way, and more generally, a molecule which is inside of
U^P-' at the time zero will be inside of Uq at p different previous times
(which will equal the negative multiples of t) .
Since E is part of U^P^ , however large p may be, as a result a mole-
cule which, at the time zero, is part of E will cross Uq an infinite
number of different times, which all equal a negative multiple of x.
There are therefore molecules which cross the volume Uq an infinite
number of times, however small this volume may be.
q.e.d.
The equations
dx ily u'o>u;>...,
and, if E has a finite volume which I may also call E, no matter what p
may be, we have E "Zi
since, without this stipulation, every molecule which was located in
this common region at the time zero would cross Oq, and consequently Uq ,
more than k times between the times -nx and 0.
We therefore have
(n-\i).
Let us now set n = k^ + x. The probability that our molecules does
not cross Uq more than k times between the times -(k^ + x)t and 0, will
be smaller than
. bV
( ^3 + a? H- I ) Uo '
It tends toward zero when k increases indefinitely.
The probability P that our molecule does not cross Uq an infinite
number of times between the times — °° and is therefore infinitely small.
In reality, this probability P is the svan of the probabilities
that the molecule crosses Uq only once, that it crosses Uq twice and
only twice, that it crosses Uq three times and only three times, etc.
However, the probability that the molecule crosses Uq k times and
k times only, between the times — «> and 0, is obviously smaller than the
probability that it will cross Ug k times or less than k times between
the times -(k^ + x)t and — it is consequently smaller than
A-V
The total probability P is therefore smaller than
V 2V kV
P<
(a- i--2)Uo (^H-9)Uo "^•■■"^ (A-'-h.rH-i)Uo
The series of the second term is uniformly convergent. Each of the
terms tends to zero when x tends to infinity. Therefore the sum of the /154
series tends to zero, and P is infinitely small.
In the same way, the probability is infinitely small that our mole-
cule does not cross Ug an infinite number of times between the times
and + «>.
The same results are obtained when any other choice is made for the
function (jj, instead of setting <|i = 1.
Equation (1) must then be replaced by the following
155
where J(oo) and JCUq) designate the integral J extended over the regions
Oq and Uq, respectively.
I shall assume that the function i|) is continuous; consequently, it
does not become infinite, and I may assign an upper limit y to it. We
then have
and since
(/iM)(<^o)<''>V,
we may deduce the following
lUV
No matter how small J(Uo) is, or how large k is, we may always take
n large enough that the second term of this inequality is also as small
as desired. We again obtain the same results which are therefor e indepen-
dent of the choice of the function <^ ,
To sum up, the molecules which cross Uq only a finite number of times
are unusual, in the same way as the commensurable numbers which are only
an exception in the series of numbers, while the incommensurable numbers
are the rule.
Therefore, if Poisson could provide an affirmative answer to the sta-
bility question which was posed, although he had excluded the cases in
which the ratio of the mean motion is commensurable, we have the right to
state that the stability which we have defined has been proven, although
we are forced to exclude the unusual molecules which we have just dis-
cussed.
I would like to add that the existence of asymptotic solutions pro- /155
vides sufficient proof for the fact that these unusual molecules exist
in reality.
Extension of the Preceding Results
297. Up to the present time, we have limited ourselves to a very
special case ~ that in which an incompressible liquid is enclosed in a
vessel, i.e.j — to employ analytical language — the case of the
156
following equations
dx __ dy dz
where X, Y, Z are three functions which are interrelated by the followinc
relationship
d\ dX d\
ax tly dz
and such that on every point of a closed surface (that of the vessel) we
have
where 1, m, n are the direction cosines of the normal to this closed
surface.
However, all of the preceding results are still valid even in the
more extended cases without changing a thing, including the line of
reasoning leading to these results.
Let the n variables xi, xg, ..., x^, satisfy the differential equa-
„, i'fi _ -A. (4)
We are therefore led to compile the following curves
V-)- "-(^'HTi^)-,. const.
The first term in relationship (4) is necessarily positive, because
we have
V — — -1- — I
where m and m are the masses of the two principal bodies, and r^ and v^
are their distances to the zero mass. The first term of (4) becomes in-
finite for r =0, for r = 0, as well as at infinity. It must therefore
159
have at least a minimum, and two points where its two first derivatives
vanish without there being a maximum or a minimina.
More generally, if there are n relative minima or maxima, there
will be n + 1 points where the two derivatives vanish without there being
a maximum or a minimum.
However, it is apparent that these points, where the two derivatives
vanish, correspond to the special solutions of the three-body problem
which Laplace studied in Chapter VI of Book X of his Mecanique Celeste
(Celestial Mechanics) .
Two of these points may be obtained by constructing an equilateral
triangle on m]^m2, either above or below the line mim2 which we shall use
for the axis of the 5's. The third apex of this triangle represents one
of the solutions in question.
All the other points satisfying the condition are located on the axis
of the C's. It may be readily seen that the first term of (4) has three
minima, and only three minima, when E, varies between — «> and +». The
first minimum is located between infinity and the mass m^, the second is
located between the two masses m^ and m2, and the third is located between
infinity and the mass m2 .
dV 9
The derivative — + n'^E, only vanishes (for n = 0) once in each of
these intervals, since it is the sum of three terms which all increase.
The equations /159
dV _ dV
— TJ- -t- /2 - f = — h «' 7) — O
indicating that the first derivatives of the first term of (4) are zero,
have only five solutions, namely, the points Bj and B2 which are the
apexes of the equilateral triangles, and the points Ai , A2 and A3 located
on the axis of the 5's. We shall assume that these points occur in the
following order
— x), A|, m,, A,, m,, A3, -H ».
We must now determine which of these points correspond to a minimum,
and we know in advance that there are two.
We should point out that if we vary the two masses mj and m2 continu-
ously, any of the five points A and B will always correspond to a minimum,
or will never correspond to one. One may only proceed from one case to
another if tjie Hessian of the first term of (4) vanishes, i.e., if two of
160
the points A and B coincide, which will never occur.
It is sufficient to examine a special case — for example, that
in which m^ = m2. In this case, the S3nmnetry is sufficient for indi-
cating to us that the two solutions A^ and A3 must have the same nature,
just like the two solutions Bi and B2. It is therefore Ai and A3 alone,
or Bx and B2 alone, which correspond to a minimum. Therefore, A does
not correspond to a minimum.
It can be seen that A^ does not correspond to a minimum.
The two minima correspond therefore to Bj and B2.
Let us now assume that mi is a great deal smaller than m2 , which is
the case in nature.
For sufficiently large values of -h, the curve
will be composed of three closed branches Cj encircling mx , C2 encircling
m2, and C3 encircling Ci and C2 . For smaller values, it will be composed
of two closed branches, Cj encircling mx and m2 , and C2 encircling Cx-
For values which are still smaller, we shall have only one closed
branch leaving mx and m2 on the outside, and encircling Bx and B2.
Finally, for even still smaller values, we shall have two closed /160
symmetrical curves, each of which encircles Bx and B2, respectively.
The statements below will only apply to the two first cases; we
shall therefore put the last two cases aside.
In the first case, the group of points satisfying the inequality (4)
may be divided into three partial groups: The group of points which are
inside of Cx , the group of points which are inside of C2 , and the group
of points which are outside of C3.
In the second case, the group of points satisfying (4) may be
divided into two partial groups : The group of points which are inside
of Cx , and the group of points which are outside of C2.
The statements below do not apply either In the first case to the
group of points which are outside of C3, nor in the second case to the
group of points which are outside of €2-
On the contrary, in the first case this applies to the group of
161
points which are inside of Ci or to the group of points which are in-
side of C.2 and, in the second case, to the group of points which are
inside of Cj .
In order to formulate these ideas more clearly, let us consider
the first case and the group of points which are inside of C2.
As the region V we shall take the region defined by the inequalities
f- -t- t'' n'
2 X (5)
We shall assume that e is small and that h has a value which we
have employed in the first case. Finally, in order to conclude the
definition of the region V, we shall impose the condition that the point
(C, n) is located within the curve C2.
It is then clear that, if the point (5, n, 5', n') is located in the
region V at the initial instant of time, it will always remain there.
In order to illustrate the fact that the results presented in the
preceding paragraphs may be applied to the case which we are discussing,
we must now show that the integral
fdlclrid'-'dr/ (2)
extended over the region V is finite.
How may this integral become infinite? Due to the fact that the /161
curve C2 is closed, E, and n are limited. The integral can therefore
only become infinite if C and n' are infinite. However, because of
the inequalities (5) , ^ and n' may only become infinite if
becomes infinite, or — since E, and n are limited — if V becomes infinite.
However, V becomes infinite for rj = and for r2 = 0, Since the
point mj is outside of C2, we need only examine the case of
Let us therefore evaluate the portion of the integral which is in
the vicinity of the point m2. If r2 is very small, £;^ + n^ is equal to
(0 m2) , and. the term — is also constant, so that if we set
162
H will be regarded as a constant.
If we then set
(5 — 0//!j) = Tjcosu, 1!) = Tj sino); 5'=pcosp;_^>„_, (5.)
and the integral (2) will become
I pr^dp d/'i dt.0 df. \^ )
We shall add the inequality
to the inequalities (5'), where a is very small, since it is the part
of the integral which is close to m2 which must be evaluated, and since
the other part is definitely finite.
If we integrate with respect to co and (j), the integral (2') will be-
come
4-^' / pr^ap dr,.
(2")
Let us integrate first with respect to p. We must calculate the /162
integral
which is chosen between the limits
/.(ll-..-^') and Py.(H^e-.^),
which provides us with e.
The integral (2") may therefore be reduced to
163
\^
/•j dri
It is therefore finite.
The theorems which were proven above may be applied to the case
which we are discussing. The zero mass will repass its initial position
as close as may be desired an infinite number of times , if one does not
impose certain unusual, initial conditions for which the probability
is infinitely small.
In the restricted problem, if we assume that the initial conditions
are such that the point E,, n must remain within a closed curve Ci or
C2, the first of the stability conditions, which were defined in No. 290,
is fulfilled.
However, the third condition is also fulfilled; therefore, Poisson
stability exists.
300. The result will clearly be the same whatever the law of
attraction may be.
If the motion of a material point 5, n is governed by the equa-
tions
d^^ _ dV J^ _ dV
dO~ ~d\' ~dif " d^
or, in the case of relative motion, by the equations
^»? dn
dC^ *" dt =
rfV
dV
= d^'
in such a way that the energy integral may be written /163
and if the function V and the constant h are such that the values of 5
and of n remain limited, we shall have Poisson stability.
However, this is not all. The same holds true in the more extended
case.
Let xi, X2, . . . , Xjj be the coordinates of -r material points.
164
Let V be the force potential depending on these n variables.
Let mj , m2 , . . . , nijj be the corresponding masses , in such a way that
we may employ m^ , m2 or m3 at random to designate the mass of the material
point whose coordinates are xi, X2 and X3.
The equations may be written
d^T-i _ dV
"'' ^ii' - dFt
and the energy integral may be written
In virtue of this equation, if the function V and the constant h
are such that the coordinates x^ are limited, there will be Poisson
stability.
What must be demonstrated is the fact that the integral invariant
Jdx', dx\ . . . dx\ dxsdxt... dx„ (x'i =~)
is finite when the integration is extended over the region I have called
V, which is defined by the inequalities
Let us call A the integral
jdx\dx\...dx'„,
extended over the region defined by the inequality
The same integral extended over the region /164
will obviously be
AR".
165
When extended over the region defined by the inequalities (1) , it will
be
or, since e is very small.
Our integral invariant therefore equals
n A ij( V + hy~'d.T, dx,... dx„ ^2)
and the integration must be extended over every point, such that V + h
is positive.
According to my hypothesis, the region V + h > is limited.
It may then be readily verified whether the integral (2) is finite
or infinite.
It will always be finite if n = 2, because the exponent of V + h is
then zero.
Let us now assume that n is > 2, and that V + h becomes infinitely
large of the order p when the distance between the two points xj , X2, X3
and X[+, X5, xg becomes infinitely small of the first order.
Then the quantity under the sign / in the integral (2) is of the
order
'(':-)■
The subset
has n - 3 dimensions. The integral is of the order n; the condition under
which the integral is finite may therefore be written
/i-(n -3)>^(^^-,j,
from which it follows that /165
^<^-r
166
This is the condition under which there is Poisson stability.
Application to the Three-Body Problem
301. The preceding considerations apply to the case in which the
following equation
/.I ■>. \ ill J
results in the fact that the x^'s can only vary between finite limits.
Unfortunately, this is not the case in the three-body problem. I
shall employ the notation presented in No. 11. I shall use x^, X2, X3
to designate the coordinates of the second body with respect to the first,
xi+, X5, xg to designate the coordinates of the third body with respect to
the center of gravity of the first two, a, b, c to designate the dis-
tances of the three bodies, and Mi, M2 , M3 to designate their masses.
Finally, I shall employ
mi — nil ~ '"3 ~ ?>
in; — m-^ -- /Hi -- ^'
to designate the quantities which I have called 6 and 3' in No. 11.
We shall then have
a ' b c
Equation (1) entails the inequality
V ; A>o. (2)
The function V is essentially positive. Therefore, if the constant
h is positive, the inequality will always be satisfied. However, the
question is whether we may assign small enough negative values to h so
that the inequality can only be satisfied for limited values of the
coordinates x^. This amounts to inquiring whether the inequality /166
^h^^^^^'!l^^.^h>o (3)
a c
with those which are imposed at the three sides of a triangle
a-hb>c, b + c>a, rt + c>6 (^)
can only be satisfied for finite values of a, b, c. Let us set a = c,
and assume that it is very large; we shall assume that b is very small.
167
The inequalities (4) will be satisfied by them.
With respect to inequality (3) which becomes
a b
no matter what h may be, it may be satisfied by arbitrarily large values
of a.
No matter how small h may be, or how large a may be, we may always
assume that b is small enough that the first term may be positive.
The existence of area integrals does not modify this conclusion.
These integrals may be written
I P(:r,37', — a-|a7'3)-H,3'(.ro^; — ar^x',)^ «,, (5)
' ^'^^\x\ — Xix\)-\-^'\x^x\ — XiX\')^. aj.
In virtue of these equations , we have
2k' + x- + .-)+|'(x- + x- + ..v)> "L±|l±_"J, (6)
where I is the moment of inertia of a system which is formed of
two material points whose masses are B and 3' and the coordinates
with respect to three fixed axes are xi , X2, X3; x^, X5, xg.
I repeat, that I is the moment of inertia which this system would have
with respect to the line serving as the instantaneous axis of rotation
for a solid, which would coincide momentarily with this system and
would rotate in such a way that the area constants are the same as
for the system.
Inequality (2) must then be replaced by the following
V + A>°? + '^|_J1^. (2')
/167
However, this equality, just like inequality (2) itself, may be satisfied
by arbitrarily large values of the x^'s, because — for very large values
of the x^'s — the moment of inertia I is very large, and, due to the fact
that the second term is very close to zero, we return to inequality (2).
We must therefore conclude that the considerations given in the
preceding section are not applicable.
In order to provide a better determination of this, let us calcu-
late the integral invariant
168
/ d.v\ dx\
. . dx'^ dxi cfxi . . . dXi,
extending it over a region defined by the following inequalities
k — e o
Is infinite, although the denominator VI1I2I3 becomes infinite when one
of the points XI, X2, X3 or x^, X5, xg recedes indefinitely. The inte- /168
gration field is then triply infinite, and the denominator only becomes
doubly infinite.
302. Even if the considerations presented in the preceding sections
are no longer applicable, we may nevertheless draw certain interesting
conclusions from the existence of the integral invariant.
Let us assume that the distance b of two of the bodies becomes small,
and that the third body recedes indefinitely. Due to its great distance,
the third body will no longer disturb the motion of the first two, which
will become essentially elliptic.
169
This third body will essentially describe a hyperbola around the
center of gravity of the first two.
In order to elucidate this point, I shall present a simple example.
I shall assume that we have a body describing a hyperbola around a
fixed point. The hyperbola is composed of two branches. One of these
branches is the analytical extension of the other, although the tra-
jectory is only composed of one single branch for the engineer.
We may then inquire whether the trajectory has an analyti-
cal extension in the case of the three-body problem, and how it may be
defined.
The coordinates of the second body with respect to the first are
xi , X2, X3; the coordinates of the third body with respect to the center
of gravity of the first two are x^, X5, xg , so that we must envisage the
motion of two imaginary points whose coordinates , with respect to three
fixed axes, are xj , X2 , X3 for the first and x^, X5, X5 for the second.
The first of these points will essentially describe an ellipse, the
second essentially a hyperbola, and it will continue receding indefin-
itely on one of the brances of this hyperbola. In order to obtain the
desired analytical extension, let us construct the second branch of this
hyperbola, and let us relate it to the ellipse described by the first
point .
Let us then consider two special trajectories of our system. For the
first, the initial conditions of motion will be such that, if t is positive
and very large, the point x^, X5, X5 will be very close to the first branch
of the hyperbola and the point xi, X2, X3 will be very close to the /169
ellipse, in such a way that the distances of these tvro points — either
to the hyperbola or to the ellipse — tend to zero when p increases
indefinitely.
Let us take the asymptote of the hyperbola as the axis of the xi+'s,
and let V be the velocity of the point which describes this hyperbola,
for a value of t which is positive and very large. Then
Xi — yt
will tend toward a finite and determinate limit X when t increases in-
definitely.
In the same way, let n be the mean motion on the ellipse and £ be
the mean anomaly, and the difference
I — nt
170
will tend toward a finite and determinate limit Iq.
If we specify the ellipse and the hyperbola and, consequently, V and
n, and in addition if we specify X and Iq, the initial conditions of motion
corresponding to the first trajectory will be completely determined.
Let us now consider the second trajectory, and let us assume that
the initial conditions of motion are such that, for t which is negative
and very large, the point xit, X5, xg Is very close to the second branch
of the hyperbola, and the point xi , X2, X3 is very close to the ellipse,
and that these two points come together indefinitely from these two
curves when t tends toward — <».
The differences
n- V/, I — lit
tend toward the finite and determinate limits X' and £'0 when t tends
toward infinity.
The initial conditions corresponding to the second trajectory are
completely defined when we specify the ellipse, the hyperbola, and X' and
If we have
the two trajectories may be regarded as the analytical extension of each
other .
Let us now consider a system of differential equations /170
'1^1 = S (,■.= ,,,., ...,rt), (1)
dt
where the functions Xj^, which depend solely on xi, X2, ..•, x^, satisfy
the relationship
Y dXi _ ^
These equations will have the integral invariant
r (2)
Let us assume that we know arbitrarily that the point x^ , X2 Xn
must remain within a certain region V, which is similar to the region V
which was considered in the preceding sections, but extending indefinitely
171
so that the integral (2) extended over this region is infinite. The
conclusions of Nos . 297 and 298 will no longer be applicable.
However, let us replace equations (1) by the following
where M is a given arbitrary function of xi, X2, .... x^^. The point
xi , X2, ..., x^, whose motion is defined by equations (1'), will describe
the same trajectories as that whose motion is defined by equations (1).
The differential equations of these trajectories are in both cases
dx\ __ dx, _ dj-^
X, " X. -• •= XT'
However, if I employ P to designate the point whose motion is de-
fined by equations (1) and P' to designate that whose motion is defined
by equations (1'), we may see that these two points describe the same
trajectory, but obey different laws.
If I employ t to designate the time when P passes by a point of its
trajectory, and t' to designate the time when P" passes by this same point,
these two times will be related in the following way
dt _ I
dt' " ^i '
We have /171
which indicates that the equations
a I
have the integral invariant
J^ldx,dTt...dx„. (2')
Let us assume that the function M is always positive, and that it
tends toward zero when the point x^, X2, . . . , Xn recedes indefinitely,
and recedes rapidl y enough that the integral (2') extended over the region
V is finite .
The conclusions presented in Nos. 297 on may be applied to equations
(1'). Tnese equations (1') therefore have Poisson stability. Since they
define the same trajectories as equations (1), it may be stated in a
172
certain sense that the trajectories of the point P also have Poisson
stability.
I shall clarify this point.
We have
i^ f''^. (3)
Since M is essentially positive, t increases with t' . However, since M
may vanish, it may happen that the integral of the second term of (3)
is infinite.
For example, let us assume .that M vanishes for t' = T ; then t will
be infinite for
r ■- T or for «'>t.
Let us consider the trajectory of the point P' . We may divide it
into two parts, the first which P' traverses from the time t' = to the
time t'= T; the second C" which P' traverses from the time t' = T to
t' = CO .
The point P will describe the same trajectory as P', but it will only
describe the part C, because it can only reach the part C' after an in-
finite time t.
For the engineer, the trajectory of P would only be composed of C.
For the analyst, it would be composed not only of C, but also of C , /172
which is the analytical continuation .
Let us imagine a point Pj whose position is defined as follows: The
point Pi will occupy at the time t^ the same position that the point P'
occupies at the time t' . With respect to tj, it will be defined by the
equality
'.- r'^^' (where i'„>T).
J,, M
The motion of the point Pj will conform to equations (1) , and this
point Pi will describe C , in such a way that the trajectories of the
points P and P^ may be regarded as the analytical continuation of each
other.
Let us now assume that the point P is within a certain region Ug at
the initial instant of time. If the initial conditions of motion are not
unusual, in the sense attributed to this word in No. 296, the trajectory
173
of the point P and its successive analytical continuations will cut
across the region U an infinite number of times, no matter how small
it may be. However, it may happen that the point P never re-enters
this region, because this region is not traversed by the trajectory,
strictly speaking, of the point P, but by its analytical continuations,
303. This may be applied to the three-body problem.
We saw above that we must consider the integral
/ rfrj . . . (Imc dx\ . . . dx\,
which we have reduced to the sixfold integral
/(v-'.--^!-^)
a J \ ' dXi dr, . . . dx^
/'>I.I.
However, we have seen that this integral, extended over the region V,
is infinite, and this has prevented us from arriving at Poisson stability.
Let us write the equations of motion in the form /173
dxi _ d.c'i
"di -' ^'' 'Ji ^ ^''
where the Xj^'s and the Y^'s are functions of the xi's and the x'.'s.
Then let us set
M — '
'^ (x]-T-xl-^xl-i-.. .-hxl-i-i)*
and let us write the new equations
d.Vi ^ X,- dx'i _ Y,-
dc' "" M ' d7 " M ■
The new equations will all have the following as the integral invariant
/ .M dXi . . . dXi dx\ . . . dx'g
or
However, this integral is finite .
Therefore, if the initial situation of the system is such that the
point P in space has 12 dimensions whose coordinates are
174
a;,, .r,, ..., .v^, j:,, x,, ..., a-;,
and if this point P is within a certain region Uq at the initial in-
stant of time, the trajectory of this point and its analytical continua-
tions — such as we have defined at the end of No. 302 — will cut
across this region Uq an infinite number of times unless the Initial
situation of the system is not unusual, in the meaning attributed to
this word in No. 296.
304. It may first appear that this result is only of interest for
the analyst, and has no physical significance. However, this point of
view is not entirely justified.
It may be concluded that, if the system does not repass arbitrarily
close to its initial position an infinite number of times, the integral /174
/
Jt
{.t] + x] +. . .H-.rJ-hl)'
will be finite.
This proposition is valid, if we overlook certain unusual trajec-
tories whose probability is zero, in the meaning attributed to this
word in No. 296.
If this integral is finite, it may be concluded that the time during
which the perimeter of the triangle formed by the three bodies remains
less than a given quantity is always finite.
175
CHAPTER XXVII
THEORY OF CONSEQUENTS
305. We may obtain other conclusions from the theory of inte- /175
gral invariants which will be of use to us below, although they will
be presented in a somewhat different form.
Let us commence by investigating a simple example. Let us assume
a point whose coordinates in space are x, y and z and whose motion is
defined by the equations
>'■' ^ V ''y -y 'i'- . z. (1)
-cu--^' di '^' di
where X, Y and Z are the given, uniform functions of x, y, z. Let us
assume that X and Y vanish all along the z axis, in such a way that
X --. y -^ o
is a solution of equations (1) .
Let us then set
a: = p coso), y ~ ? sinu),
and equations (1) will become
§ = «. S-". S=^. <^'
where R, Q, and Z are the functions of p , w and z which are periodic
having the period Its with respect to oj.
It is advantageous for us to assign only positive values to p, and
we may do this with no difficulty since x = y = is a solution.
I shall now assume in addition that n can never vanish and, for
example, always remains positive. Then u will always increase with t.
Let us assume that equations (2) have been integrated, and that we
have the solution in the following form /176
The letters a and b represent integration constants.
Let us set
176
Po--^/ifo, (i,b), Zt,rr-.f^{o, a, b),
Pi --^/i(2-, 'T, b), z; =--f.i(?.r., a, b).
Let Mq be the point whose coordinates are
■^ " ?<>• y - 1>. " .= -0.
and Mj be the point whose coordinates are
These two points both belong to the half-plane of the xz's located
on the side of the positive x's.
The point M^ will be the consequent of Mq.
If we consider the bundle of curves which satisfy the differential
equations (1), if we pass a curve through the point Mq , and if we extend
it until it encounters the half-plane (y = 0, x > 0) again, the preceding
definition is justified by the fact that this new encounter will occur at
Ml.
If an arbitrary figure Fq is drawn in this half-plane, the conse-
quents of the different points of Fq will form a figure Fi which will be
called the consequent of Fq.
It is evident that pi and z^ are continuous functions of Po and
Zq.
Therefore, the consequent of a continuous curve will be a continuous
curve, the consequent of a closed curve will be a closed curve, and the
consequent of an area which is connected n times will be an area which
is connected n times .
Let us now assume that the three functions X, Y and Z are related as
follows
^MX ^MY dm. _
dx dy dz ^ '
where M is a positive, uniform function of x, y, z.
Equations (1) then have the integral invariant
J Mdirdy dz
m
and equations (2) have the following invariant ,,^^
/ M p dp (/w (/;;.
Let us now consider the equations
'II ^ R ^'^ _ z Jz
(see No. 253) .
/ MO.o dp o),
and let Fi be its consequent.
Let Jq be the Integral
hlQpdpd::, (5)
extended over the planar area Fg , and let Jj be the same integral ex-
tended over the planar area Fj .
Then let $o ^^ the volume produced by the area Fq when it is ro-
tated around the z axis by an infinitely small angle e, and the in-
tegral (4) extended over $ will be J e.
^
In the same way, let $i be the volume produced by the area F^ when
it is turned around the z axis by an angle e, and the integral (4) ex-
tended over $ will be J. e.
1 1
The integral invariant (4) must have the same value for $o as for
$1, and we must have
Jo ^ - Jl .
Thus, the Integral (5) has the same value for an arbitrary area and
its consequent .
178
This is a new form of the basic property of integral invariants.
306. Let us then assume a closed curve Cq located in the JJJA
half-plane (y = 0, x > 0) and encompassing an area Fq. Let Ci be
the consequent of Cq . This will also be a closed curve which will en-
compass an area Fi , and this area Fi will be the consequent of Fq.
If the integral (5) , extended over Fq and over Fi , has the value
Jo and Ji , we shall have
Jo-- J 1 I
from which it follows that Fq cannot be a part of Y^ , and F^ cannot
be a part of F.,
Four hypotheses may be formulated regarding the relative position
of the two closed curves Cq and C^.
1. Ci is within Cq;
2. Co is within Ci ;
3. The two curves are outside of each other;
4. The two curves intersect.
The equation Jq = Ji excludes the two first hypotheses.
If the third is also excluded, for whatever reason, the two curves
will definitely intersect.
For example, let us assume that X, Y, Z depend on an arbitrary
parameter y and that for p = 0, Co is its own consequent. For very small
values of u, Co will differ very little from Ci. Therefore, it could
not happen that the two curves Cq and Ci are outside of each other, and
they must intersect.
Invariant Curves
307. Any curve which will be its own consequent will be called an
invariant curve .
Invariant curves may be readily formed. Let Mq be an arbitrary
point of the half-plane, and let M^ be its consequent. Let us connect
Mo to Ml by an arc of an arbitrary curve Cq. Let Ci be the consequent
of Co, C2 be the consequent of Cj, and so on. The entire group of arcs
of the curve Cq, C^ , C2 , ... will obviously constitute an invariant
curve .
179
But we may also consider Invariant curves whose formation will be
more natural.
Let us assume that equations (1) have a periodic solution. Let /179
^ = ?i(0, r = ?!(0, - = =3(0 ^^^
be the equations of this periodic solution, in such a way that the func-
tions (i>^ are periodic in t, having the period T.
I shall assume that when t increases by T, oj increases by 2tt .
Equations (6) represent a curve. Let Mq be the point where this
curve intersects the half-plane; this point Mq will obviously be its own
consequent .
Let us now assume that there are asymptotic solutions which are very
close to the periodic solution (6) . Let
^ = *.(0> 7 = *2(0. z^'P,{i) (7)
be the equations of these solutions.
The functions $£ may be developed in powers of Ae^'^, and the co-
efficients are themselves periodic functions of t. In this expression,
a is a characteristic exponent, and A is an integration constant.
In equations (7) , the three coordinates x, y, z are therefore ex-
pressed as a function of two parameters, A and t. These equations there-
fore represent a surface which may be called the asymptotic surface .
This asymptotic surface will pass through the curve (6) , since equations
(7) may be reduced to equations (6) when we set A = 0.
The asymptotic surface will intersect the half-plane along a
certain curve which passes through the point Mq and which is obviously
an invariant curve.
308. Let us consider an invariant curve K. I shall assume that X,
Y, Z depend on the parameter y, as well as the curve K.
I shall assume that for y = 0, the curve K is closed, but that it
ceases to be closed for small values of y.
Let Aq be a point of K. The position of this point will depend on
y. For y = 0, the curve K is closed, so that, after having traversed
this curve starting with Aq , one returns to the point Ag . If y is very
small, this will no longer be the case, but one will pass very close to
Aq. Therefore, on the curve K there will be a curve arc which is
180
different from that where Aq is located, but which will pass very close
to Aq. Let Bo be the point of this curve arc which is closest to Aq. /180
I shall join AqBq.
Let Ai and B^ be the consequents of Aq and Bq. These two points
will be located on K. Let AiBj be the consequent curve of the small
line AqBo-
We must consider the closed curve Cq which is composed of the arc
AqMBq of curve K, included between Aq and Bq, and of the small line
AqEq. What will its consequent be?
In order to define our ideas more precisely, let us assume that
the four points Ai , Aq , B^, Bq follow each other on K in the order
AiAqBiBq.
The consequent Ci of Cq will be composed of the arc AiMBi of the
curve K and of the small arc AiBi, the consequent of the small line AoBq.
Several hypotheses may then be formulated:
1. The small curvilinear quadrilateral AqBoAiBi is convex, that is,
none of these curvilinear sides have a double point, and the only points
which the two sides have in common are the apexes. In this hypothesis,
the form of the curve would be that indicated in one of the following
figures
Figure 1 Figure 2
This hypothesis must be rejected, because it is apparent that the
integral J ig larger in the case of Figure 1 for Ci than for Cq, and
smaller in the case of Figure 2.
181
2. The arc AqAi or BqBi has a double point. If this were the
case for the invariant curve K, there would have to be a double point on
the arc joining an arbitrary point on the curve to its first consequent; /181
we shall assume that this is not the case. Actually, this condition
would not occur in any of the applications which I have in mind. It does
not apply, in particular, in the case of the invariant curve produced by
an asymptotic surface , as I explained at the end of the preceding section.
It may be readily stated that the asymptotic surface does not have a
double line if we limit ourselves to the portion of this surface corres-
ponding to small values of the quantities which I have designated as
Ae'^^ above.
On the other hand, the line AqB does not have a double point,
and the same must be true for its consequent A^Bx. To sum up, we shall
assimie that the four sides of our quadrilateral do not have a double
point.
3. The arc AqAj intersects the arc BoBj. (As a special case, this
case includes that in which the curve K would be closed.) Our curves
will then have the form shown in Figure 3.
Figure 3
4. The arc AqEq intersects its consequent AjBi. Our curves will
then have the form shown in Figure 4.
There are cases in which this hypothesis must be rejected. For
example, let us assume that X, Y, Z depend on one parameter y, and that
for V = the curve K is closed and that each of its points is its own
consequent, so that for y = the four apexes of the quadrilateral
coincide.
Then the four distances AqBq, A^Bi, AjAq, BiBq will be infinitely
small quantities if y is the main infinitely small quantity. Let us /182
assume that AjAq is an infinitely small quantity of the order p,
AqBq an infinitely small quantity of the order q, and that q is
182
Figure 4
larger than p.
Since A^Bi is the consequent of AgBo, the length of the arc A^Bi
must be of the order q. Then let C be one of the intersection points
of AgBQ. In the mixtilinear triangle whose two sides are the lines
AiAq and AqC, and whose third side is the arc of the curve A^C which is
part of AiBi, the side AjC is larger than the difference between the two
others, it should therefore be of the order p, and we have seen that it
must be of the order q.
The hypothesis must therefore be rejected.
AiAg
5. Two adjacent sides of the quadrilateral intersect, for example ,
and AjBi. It is then necessary that AqBq, which is the antecedent
of AiBi, intersect K itself. If A'q is the intersection of AqEq with K,
and A'l is the intersection of A^Bi with the arc AqAj , A'l will be the con-
sequent of A'o, and we shall obtain the following figure.
Figure 5
It is apparent that A'q and A\ may play the same role as Aq and Ax,
183
and that we therefore return to the first case.
This new hypothesis must therefore be rejected. J183
To sum up, the two arcs AqAj and BqBi will intersect every time that
hypotheses 2 and 4 must be rejected, for one reason or another.
We must now examine the case in which the points Ai , Aq, Bi, Bq
follow one another in a different order on K. The orders BiBqAiAq,
BoBiAqAi, AqAiBoBi do not differ essentially from that which we have
just studied.
Orders such as AiBiBqAo, AiBqBiAo, AiBqAqBi, ... will not appear
in the applications which follow. We shall always assume that, if y is
very small, the distances AqAi and BqBi are very small with respect to
the length of the arcs AqMBq or A^MBi.
The order AiAqBqBi, or the equivalent orders, remain, and we shall
no longer discuss them. It is apparent that if they appear, on the arc
AqMBq there will be a point which will be its own consequent.
309. For example, let us assume that equations (1) have a periodic
solution
^ = ?.(0. 7 = ?5(0, •5 = ?j(0 (6)
and asymptotic solutions
a: = *,(0, 7 = *5(0, z = ^,{t). (7)
Let us assume that equations (1) depend on a very small parameter
y, and that X, Y, Z may be developed in powers of this parameter.
For y = 0, let us assume that the asymptotic solutions (7) may
be reduced to periodic solutions. This may be done as follows. We
have stated that the $i's may be developed in powers of Ae'^ , with the
coefficients themselves being periodic functions of p. However, the
exponent a depends on y; let us assume that it vanishes for y = 0.
Then for y = the functions ^^ will become periodic functions of t,
and the solutions (7) may be reduced to periodic solutions.
The asymptotic surface intersects the half-plane along a certain
curve Co which passes through the point Mq, which is the intersection of
the half-plane with the left curve (6) . 11^
The curve Cq is obviously invariant, as I stated at the end of
No. 307. Foy y = 0, each of the points of Cq is its own consequent.
184
In addition, I shall assvime that the curve Cq is closed for
p = 0.
Let us refer back to Chapter VII, Volume I. We saw from Nos. 107
on that, in the case of dynamics, the characteristic exponents may be
developed in powers of /y", and are equal pairwise and have the opposite
sign. We shall assume that this is the case.
In reality, we then have two asymptotic surfaces corresponding to
the two equal exponents having opposite sign a and -a. We therefore
have two curves Cq which will intersect at the point Mq.
We may distinguish between four branches of the curve
c c c r"
all four of which end at the point Mq ; Cq and Cq will correspond to
the exponent a, C'^ and C" to the exponent -a.
Figure 6
These different branches of the curve are shown in Figure 6. The
branch C'q is the branch MoPoPiAqAi, the branch Cq is the branch MqEqEi,
the branch C^ is the branch MqQiQo and the branch C" is the branch
MoRiRoBiBq.
These four branches of the curve are obviously invariant.
Now, for y = 0, C'q is identical to C^ , Cq is identical to C'j", and
(if we assume that the curve Cq is closed for y = 0. which we shall /185
call cj,) these four branches of the curve will coincide on the closed
curve Cq.
It may be deduced from this that, for very small y, these branches
185
of the curve will differ very little from each other, that C'g will
deviate very little from C\, CJ will deviate very little from C", and
that, if Cg is sufficiently extended, it will pass very close to C",
if it is sufficiently extended.
I have indicated on the figure different points of these branches
of the curve and their consequents. Thus, A^ , Bj, E^, P^ , Qi, Ri are,
respectively, the consequents of Aq , Bq, Eq, Pq* Qo » ^0 •
We would first like to note that the points Aj , Aq , Bj, Bq do
follow each other (as we assvmied at the beginning of No. 308) in the
order A^AqBiBo when the invariant curve formed of the two branches Cq
and C" is traversed from A], to Bq.
This invariant curve is not closed, but it differs very little from
the closed curve Cq .
In this connection, let us examine the five hypotheses of No. 308.
As we have seen, the first must be rejected. The second will no longer
occur.
It could only occur if the asymptotic surface (7) had a double
line.
We have stated that the $j^'s may be developed in powers of Ae'^'^.
Therefore let us set
,• ---. '[>? -)- Ae^''^,' 4- A'e'«''I>' -h. . . .
If our surface had a double line, this double line would have to
satisfy equations (1). Actually, the asymptotic surface is produced by
an infinite number of lines satisfying these equations in such a way
that, if two layers of this surface happen to intersect, the intersection
could only be one of these lines .
Since $-l depends on the time t and the parameter A at the same time,
we may show this by writing
'I>,-=-f,(/, A).
If there were a double line, we would have to have the three
identities
,(<, A), *,(<, A)]
and in the same way
'^*4 ==X [*,(«', B), +,(;', B), *,(<', B;],
from which it follows that
rf*, _ d'Pi dt' _
~dT ~ ~di' ' "di ~ ^'
from which we have
f=z t-hh,
where h is a constant.
We would thus obtain the following
where
G= Be^fi.
The identity must be valid for t = — <», from which it follows that
Ae»' = 065" = o,
and we have
*?(')= *?('-+- A),
from which we have h = and
or
A,'(04-... = C'J>;(0-t-C»c«'*?(0-H.-.
or, setting t = — «>, we have
A = C = B.
187
Due to the fact that the two values A and B are equal, there is no
double line. ^
The third hypothesis may be adopted.
Let us pass on to the fourth hypothesis. In order to determine
whether it must be rejected, we must try to determine the order of
magnitude of the distances AjAq and AqBo- This is what we shall do in
the different applications which follow.
Finally, the fifth hypothesis is always reduced to the first one,
as we have seen.
Extension of the Preceding Results
310. We formulated very special hypotheses above concerning equa-
tions (1), but all of them are not equally necessary.
Let us consider a region D which is simply connected and which is
part of the half-plane (y = 0, x > 0). Let us assume that we know arbi-
trarily that, if the point (x, y, z) is located at a point Mq in this
region at the initial instant of time, co will constantly increase from
to 27T when t increases from to to, in such a way that the curve satis-
fying equations (1) and passing through the point Mq ~ assuming that it
is extended from this point Mq up to its new intersection with the half-
plane — is never tangent to a plane passing through the z axis.
Just as in No. 305, we may then define the consequent of the point
Mq, and it is apparent that all the preceding statements will still be
applicable to the figures which are located within the region D.
It will not be necessary that the curves satisfying equations (1)
and intersecting the half-plane outside of D be subjected to the condi-
tion of never being tangent to a plane passing through the z axis. It
will no longer be necessary that x = y = be a solution of equations
(1).
Then, if Cg is a closed curve inside of D and if Ci is its conse-
quent, the two curves will be outside of each other or will intersect.
The results given in No. 308 will be equally applicable to the
invariant curves which do not leave the region D. If even one invariant
curve leaves the region D when it is sufficiently extended, the results
will still be applicable to the portion of this curve which is within /188
this region.
311. Let us now consider a curved surface S which is simply
188
connected, instead of a plane region D. Let us pass a curve y satis-
fying equations (1) through a point Mq of this curved surface, and let
us extend this curve until it again intersects S. The new point of
intersection Mj may still be called the consequent of Mq .
If we consider two points Mq and Mq which are very close to each
other their consequents will be, in general, very close to each other.
There would be an exception if the point Mj were located at the boundary
of S, or if the curve y touched the surface at the point M^ or at the
point Mq. Except for these exceptions, the coordinates of Mj are analy-
tic functions of the coordinates of Mg.
In order to avoid these exceptions, I shall consider a region D
which is part of S and such that the curve y, proceeding from a point
Mq inside of D, intersects S at a point M^ which is never located at
the boundary of S — so that the curve y does not touch S either at Mq
or at Mj. Finally, I shall assume that this region D is simply connected.
Let us adopt a special system of coordinates which I shall call
C, n and <;, for example, and for which I shall only assume the following:
1. When Id and | n | are smaller than 1, the rectangular coordinates
X, y and z will be analytic and uniform functions of 5, n and ?, which
are periodic with the period 27i with respect to ?.
2. No more than one system of values of 5, n, ^ can correspond to
a point (x, y, z) in space, such that
i{|(xi)
is an odd function of x^ which increases from to 1 when xj increases
from to + °o . We may set, for example
/i + ^J
If this hypothesis is adopted, the point M will always be within a torus
of radius 1, which is tangent to the z axis.
An infinite number of systems of values of x , y, and y will /191
correspond to each point M within this torus. However, these systems
will not differ essentially from each other, since one passes from one to
the other by increasing y^ or y^ by a multiple of 2ir, or by changing x,
into -X and y into y, + tt.
If xi, Yi and y2 are given, X2 may be deduced by means of equation
(2) . Let us assume that the variables x and y vary in accordance with
191
equations (1) . and the corresponding point M will describe a certain
curve which I shall call the trajectory.
One and only one trajectory passes through each point inside the
torus.
The form of these trajectories for y = may be readily determined.
For y = 0, the differential equations may be reduced to
dxi _ dy£ ^^ _
'di "" °' dt ^ dxi
The x.'s are therefore constants, which Indicates that our trajec-
tories arehocated on the tori, and the y.'s are linear functions of time,
because
rfF„
dTj
depends only on the x^'s and is a constant.
If the ratio ni :n2 is commensurable, the trajectories are closed
curves. Conversely, they are not closed if this ratio is Incommensurable,
Let mi, m2, Pi, P2 be four whole numbers, such that
Let US set
x\ = pl^i —pix„
The identity
3^'.7i -^- ^''.y'i = ^'^' "^ ^'-y'
indicates that when one passes from the variables x^ y^ to the variables
x'i, y\, the canonical form of the equations is not changed.
We shall assume that nz does not vanish when xi remains less than a
certain limit a. Then ^ will always retain the same sign, and we /192
shall have, for example rf,,.
^>'-
192
This inequality, which is valid for y = 0, will still be valid for small
values of y.
The relationships
will then define a certain plane region D which will have the form of
a circle.
The trajectories starting from a point in this region will never be
tangent to a plane passing through the z axis, at least before having
cut across the half -plane y = again. Our region may therefore play
the role of region D in No. 310.
The equations (1) have the integral Invariant
from which we may deduce the following by means of the Integral F =
const.
' A5B5
are the successive consequents of AqEq.
The arcs AiB^, A2B2, ..., A5B5 are no longer rectilinear in
general, but are very small arcs of a curve.
Figures 1 or 2 shown in No. 308 reproduce the part of the figure
shown by the solid line. The entire group of our curves shown by the
solid lines represents an invariant curve K.
I have drawn the figure based on the first hypothesis, which — as
we have seen — must be rejected along with the fifth hypothesis. Accord-
ing to the statements I made in No. 309, this also holds true for the
second hypothesis.
We must examine the fourth hypothesis In greater detail. In order
to do this, let us try to determine the equation of our asymptotic sur-
faces. Based on the statements presented in No. 207, this equation may
be obtained in the following way.
A function S is formulated which may be developed in powers of /\i,
in such a way that
p
S = SoH-/uS,-+-...-4- fi'Sp-i-....
Regarding Sp, it is a periodic funtion of the period 2Tr with
respect to y2, and Att with respect to y\.
We shall have
dy\' -^'"^
ds ds W
Equation (4) is the equation of the asympototic surface.
If the series S were convergent, the periodicity of the S 's would
entail the condition that our curves must be closed and that the two
points Aq and Bq must coincide. However, this is not the case (see No.
225 , and the following) .
What significance does equation (4) have? It may only be valid from
the formal point of view, i.e., if E is the sum of the p + 1 first /196
terms of the series S, so that
196
Sp = So+ /jJiS, -H, . .+ p' Sp,
the equation
dllp dz„
£±L
2
will be valid up to quantities of the order y
However, equation (4') represents a closed surface, and p is ar-
bitrarily large.
We must therefore conclude that the distance AqBo is an infinitely
small quantity on the order of infinity (see Nosj^ 225 on) . In addition,
the distance A0A5 (or B0B5) is on the order of /y, and is consequently
infinitely small of the order of ^.
The distance AqBq is therefore infinitely small with respect to
AqA^ , which indicates that the fourth hypothesis must be rejected.
The only possible hypothesis is therefore the third.
Therefore the two arcs A0A5 and B0B5 intersect.
Application to the Restricted Problem
313. I am going to apply the preceding principles to the problem
presented in No. 9, and I shall employ the notation given in that section.
Consequently, we shall have the canonical equations
dx\ _dF^ dy\ _ dF'
dt dy'i dt dx'i
based on which we may set
and, in addition.
X, — L, x'^ = G,
F' = R-t-G==Fo4-(jiF,4-.
(5)
IX
Let us now set
197
Xf—L — G, xj = L -r- G,
and the equations will retain the canonical form and will become /197
dxi _dr dyt ^ dr
dl ~ dyi ' dt ~' dxi
We will have
■?•; — ^1
{Xi+XtY ' 1
Fo = , ■_ — — ^
from which it follows that
+ 4 I -H 4
n, = ;--" r- -t- - , ^l =
(Xi+.r5)' ■! {Xi-'r X.,Y 2
If we assume that the eccentricity is very small, L and G will
differ very little in absolute value. Therefore, one of the two quanti-
ties xi and X2 is very small.
I would like to note in addition that the equations
L = /n, G - ^a{i^ e')
indicate that G is always smaller than L in absolute value. Therefore,
XI and X2 are essentially positive.
Let us assume that x^ is very small. The function F' will be a
function of a and of £ + g - t which may be developed in powers of
e cos g and of e sin g. Therefore, this will also be a function of X2
and of y2 which may be developed in powers of
/^ cos/i and Ai sin^,.
It will be periodic with the period 2ir both in yi and in y2.
If, on the other hand, it is X2 which is very small, the function F'
will be a function of xi and of yi , which may be developed in powers of
s/xt cosj, and /■^^i sin/i.
Let us now assxime that our four variables x and y are related by
the equation of energy
F = C .
198
This equation may be approximately reduced to
Fo = C.
Let us construct the curve Fq = C, taking x^ and X2 as the coordi-
nates of a point in a plane.
The equation may be written
This curve has two asymptotes /198
Xl -H Tj — O, a"i — T| = a C
and it is symmetrical with respect to the first of these two asymptotes.
However, it should be noted that the only portion of the curve
which is of use to us is that which is located in the first quadrant
3:1 >o, r, >o.
Based on the values of C, the curve may have one of the forms shown
in the two following figures
Figure 8
The axes of the coordinates are represented by the dot-dash line,
the asymptotes and the utilizable portions of the curve are shown by the
solid line, and the portions of the curve which are of no use are shown
by the dotted line.
199
Figure 9
We shall assume that a value Is assigned to C, so that the curve has
the form shown in Figure 9 and so that it contains two utilizable arcs
AB and CD. We shall no longer consider the arc AB.
We should point out that when one traverses this arc AB, xi de-
creases constantly from OA to zero, X2 increases constantly from zero to
OB and — increases constantly from zero to +■» .
If we now construct the curve F = C, assuming that yi and y2 are
constants and x^ and X2 are the coordinates of a point in a plane, the /199
curve will differ very little from Fq = C and can still be represented
by Figure 9. It will have a utilizable arc AB, and when one traverses
X2
this arc the ratio — will increase constantly from zero to + " .
We thus arrive at the following method of geometric representation.
The location of the system will be represented by the point whose rec-
tangular coordinates are
/^J + 4^1 — 2 /^ cos^i /xj ~^- ix, — 2 /j7, COS^i
2 /t, sinji
These three functions may be developed in powers of 1/xi cos yi and
\/k^ sin yi, if xj is very small, and may be developed in powers of
200
yx2 cos 72 and I/X2" sin 72, if X2 Is very small. They only depend on
the ratio — .
Thus, one and only one point In space corresponds to each system
of values of yi and of y2 and to each point on the utilizable arc AB.
The functional determinant of the three coordinates with respect
to yi, y2 . and with respect to ^_L , always retains the same sign.
▼ ^2
We may therefore apply the results obtained in the preceding sec- /200
tion within all of the region D where n2 does not vanish.
However, n2 vanishes for xi + X2 = 2.
But, if we have x^ + X2 = 2, x^ > 0, X2 > 0, we shall obviously
have
2 Xi — a-i ^ 1 Ti + Xi _ 3
(ri-i-.r,)» ■ Z ■" (a-,-F j-,)» 2 ~4
However, the first term of this equation is Fq and, when compiling the
curve Fq = C, we assumed that we were dealing with the case presented
in Figure 9. However, the case shown in Figure 9 assumes that
Since Fq differs very little from F, and consequently from C, we
cannot have at the same time
4 4
3
(unless C is very close to its limit -t", which we have not assumed).
Under the conditions with which we are now dealing, we shall not
have n2 = .
Thus, the results presented in the preceding section are applicable,
and if we construct the asymptotic surfaces and if we consider the inter-
section of these surfaces with the half-plane y2 = 0, the two arcs which
are similar to those which we designated as AqAs and B0B5 above will
intersect.
201
I would like to add one word to this .
The coordinates of the third body, with respect to the major axis
and the minor axis of the ellipse which it describes, are — according
to the well-known formula
L2(cos;-4-.. .),
LG{sin/ H-...).
It may thus be seen that, when G changes sign, the second of these
coordinates changes sign.
As a result, the perturbed planet turns in the same direction as the
perturbing planet if G is positive, and it turns in the opposite direc-
tion if G is negative.
202
CHAPTER XXVIII
PERIODIC SOLUTIONS OF THE SECOND TYPE
314. Let us consider a system of equations /201
W " ^' ^'"='' '' ••■' ^>' (!■)
where the X^'s are functions of x^, X2, ..., x^, and of t, which are
periodic having the period T with respect to t.
Let
(2)
be a periodic solution of period T of equations (1) .
We shall try to determine whether equations (1) have other periodic
solutions which are very close to (2) and whose period is a multiple of
T.
These solutions, if they exist, will be called periodic solutions
of the second type .
Let us consider a solution of equations (1) which is very close
to (2) . Let
9,(0) +- ?/
be the value of x^ for t = 0, and let
?<(o)- - ?i + 'I/, - o,( A-T) + p,--(- <},,■
be the value of x. for t = kT (k is a whole number) .
The B^'s and the tj^i'sj whose definition is the same as that given in
Chapter III, will be very small. Just as in Chapter III, it will be
found that the ^'s are ftinctions of the g's which may be developed in
increasing powers of the 3's.
In order that the solution may be periodic having the period kT, it /202
is necessary and sufficient that
203
Due to the fact that the (f)-j^(t)'s are periodic functions, the iJj's
vanish with the 3's.
We shall assimie that the functions X^ which appear in equations
(1) depend on a certain parameter y. Then the functions (j).(t) will
depend not only on t, but also on y. As regards t, they will be
periodic of period T, with T being a constant which is independent of
y.
Under these conditions, the functions 4), whose definition remains
the same, will depend not only on the 3's, but also on y. If we assume
that
p,, p., •••, Pi, }^
are coordinates of a point in space having n + 1 dimensions, equations
(3) will represent a curve in this space. A periodic solution, of period
kT, will correspond to each point on this curve.
Since the ((/'s all vanish when the 3's all vanish at the same time,
this curve will consist of the straight line
p,= p,-.... .;5„^e. (4)
The solution (2) will correspond to different points on this
straight line. Due to the fact that this solution is a periodic solu-
tion of period T, it is for that reason a periodic solution of period
kT.
But we must try to determine whether there are other periodic solu-
tions which are very similar to the first or — in other words — if
curve (3) includes, in addition to the straight line (4), other branches
of the curve which are very close to the straight line(4).
In other words, are there points on the straight line (4) through
which branches of the curve (3) pass, other than this line?
Let
P, == p, rr^ . . . r^ P, = O, Ji == Jio
be a point P of the line (4) .
In order that, several branches of the curve may pass through the
point P, it is necessary that at this point P the functional determinant ,
or the Jacobian, of the 4''s» with respect to the B's, vanishes .
204
This condition is not sufficient, as we shall see at a later /203
point, for several real branches of the curve to pass through the point
P.
Let us formulate the determinant of the i|;'s with respect to the
e's, let us add -S to all the diagonal terms, and let us set the de-
terminant thus obtained equal to zero. ¥e shall thus obtain the equa-
tion which is known as the equation for S .
The roots of this equation (see No. 80) are
where a is one of the characteristic exponents of equation (1) .
In order that the functional determinant may be zero, it is necessary
and sufficient that one of the roots be zero. We must therefore have
which means that kaT is a multiple of 2iir.
Therefore, in order that several branches of the curve pass through
the point P, it is necessary that one of the characteristic exponents be
a multiple of t^.
kT
315. This condition is not sufficient, and a more extensive dis-
cussion is necessary.
Let us set
and let us try to develop the g's in whole or fractional powers of X.
We shall assume that the Jacobian of the (j^'s, with respect to the
6's, is zero. This Jacobian vanishes for X = 0, but will not be identi-
cally zero, in general. In order that this may be the case, it is
necessary that one of the characteristic exponents be constant, indepen-
dent of y, and equal to a multiple of ~-.
We shall therefore assume that the Jacobian vanishes for X = Q,
but that its derivative, with respect to X, does not vanish.
In the same way, we shall assume that the minors of the first order
of this Jacobian do not all vanish at the same time.
In this case, based on the theorem in No. 30, from n - 1 of equa-
tions (3) we may derive n - 1 of the quantities 6 in the form of series
205
developed in whole powers of X and of the n^ quantity 6, for example
of 6n- ^^
Let us substitute the values of
h, p.. ■••• P"""
thus derived in the ntk equation (3). The first term of this r^
equation will be developed in powers of X and of $^. Let us write it
in the following form.
e(X, 3„) - o.
I may first point out that must be divisible by &^, because the
line (4) must be part of the curve (3).
On the other hand, the derivative of with respect to Bn '""st
vanish for X - 0, since the Jacobian vanishes. For X = 0, does not
contain a term of the first degree. Let us assume that it no longer
contains terms of the second degree. . . . . p - UiL degree, but that it
does contain a term of degree p.
Finally, since the derivative of the Jacobian with respect to X
does not vanish, we shall have a term containing Xe^-
I may therefore write
9.. A),3„-i-nii^f-C,
where C is the total group of terms containing $P , Xg^. o^ ^ &n ^^ ^
factor. A and B are constant coefficients which are not zero.
It may be seen that we may derive 6n from this in terms of a series
which progresses according to the powers of 1^' , and the problem is
to determine whether this series is real.
If p is even or if, p being odd, A and B have opposite signs, the
series is real, and periodic solutions of the second type exist.
If p is odd, and if A and B have opposite signs, the series is
imaginary, and there is no periodic solution of the second type.
I shall now assume that not only the Jacobian vanishes for X = 0,
but that the same holds true for all of its minors of the first, the
second, etc., and p - iJ^ order. I shall nevertheless assume that the /205
minors of the p£^ order are not all zero at the time.
206
According to the statements presented In No. 57, under these con-
ditions, there will be not one, but p, characteristic exponents which
Ziir
will be multiples of ttt".
^ kT
From n - p of equations (3) , we may then derive n - p of the
quantities g in the form of series developed in powers of X and of the
p last quantities B.
For purposes of brevity, I shall employ the B' 's to designate the
n - p first quantities 6, and the g" »s to designate the p last quanti-
ties 3. We shall therefore have the 3' 's developed in powers of X and
of the 3" 's .
Let us substitute these expansions in the place of the 3' 's in
the p last equations (3) , and we shall obtain p equations
e, ^ej.-^...-^ epr=o, (5)
whose first terms will be developed in powers of A and of the 3" 'g
Due to the fact that the Jacobian and its minors of the first p - 1
orders are zero, these first terms will not include terms of the first
degree in 3" which are independent of X. We must now determine whether
the first terms of equations (5) will contain terms of the first degree
with respect to the 3" 's, and at the same time of the first degree with
respect to X.
Let 6-^ be the total group of terms of 0^ which are of the first
degree with respect to the 3" 's- It is apparent that 6^ may be developed
in powers of A. Let
be this expansion. The e|^)'s will be homogeneous polynomials of the
first degree with respect the the 3" 's.
According to the preceding statements, e9 will be identically zero,
but we must now determine whether the same holds true for q\ .
The Jacobian of the ip's with respect to the 3's equals
n(i --e'-^'f),
The product indicated by the sign IT extends over n factors corresponding
to the n characteristic exponents a.
207
Let ai , 02, ..., o^ be these n exponents, and let /206
(j> (a )-0 — £*'■•■)
The Jacobian will equal the product
?(ai)?(aj)--.'f(i/i).
In order that A = 0, the Jacobian vanishes as well as its minors
of p - 1 first orders. As a result, p of the exponents are multiples
of -j-^. Therefore, p of the factors ())(oi) vanish for X = and are,
consequently, divisible by X. The product, i.e., the Jacobian, will
therefore be divisible by XP.
dct
We shall assume that for X = none of the -r- vanishes, which is
dX
what we already assumed previously. Under these conditions, none of
the (j)(a)'s are divisible by X^. Therefore, the product is not divisible
by XP+1.
Thus, the Jacobian is divisible by XP , but not by xP
As a result, the determinant of the 6, 's is different from zero,
and consequently none of the O.'s vanishes identically.
The simplest case is that in which, for X = 0, the terms of the
second degree do not vanish in the expressions for 0^, and in which
these terms of the second degree cannot vanish at the same time, unless
all the 3" 's vanish at the same time.
Let us assume that m is the total group of terms of the second
degree of 0-j^ for X = 0.
It will be sufficient to consider the algebraic equations
T„+X8,':---0,
whose first terms are homogeneous polynomials of the second degree with
respect to X and the g" 's.
If these equations have real solutions, we shall have periodic
solutions of the second type.
I shall not extend the discussion to the other cases, but shall com-
plete this discussion when treating the equations of dynamics.
208
Case In Which Time Does Not Enter Explicitly 1201
316. Let us assume that the functions X^ which appear in equations
(1) do not depend on time t.
As we have seen in No. 61, in this case one of the characteristic
exponents is always zero.
In addition, if
is a periodic solution of period T, the same also holds for
whatever the constant h may be.
In the preceding section, we assumed that — no matter what y
might be — there was a periodic solution
and the period could only be T, since the Xj^'s were periodic functions
of t, of period T.
The period was therefore independent of \i.
The same is not true in this case. We shall always assume that,
no matter what y might be, equations (1) have a periodic solution
However, the period will depend on y, in general. I shall call
T the period, and Tg the value of T for y = yg, i.e., for X = 0. We
shall then modify the definition of the quantities g and (j; to a certain
extent .
We shall always designate the value of x^ by ^±(0) + 6^ for t = 0,
However, we shall designate the value of x^ by (|>j^(0) + g^ + tj^^^ for
t = k (T + x) (and not for t = kT) .
Then, the 4i-l's will be functions of the n + 2 variables
209
If we continue to assume that the g's and A's are the coordinates
of a point in space having n + 1 dimensions, the equations /208
:• o (3)
will no longer represent a curve, but will represent a surface, since we
may vary the two parameters t and X independently and continuously.
However, we should point out that curves are drawn on this surface
whose different points correspond to periodic solutions which may not be
regarded as being essentially different.
If
is a periodic solution, the same will hold true for
no matter what the constant h may be, and this new solution will not differ
from the first in reality.
The following point corresponds to the first
p,-.=/,(o) -0,(0),
and the following point corresponds to the second
?<■ -/,(/') -?/(o).
When h is varied continuously, the second point describes a curve whose
different points do not correspond to solutions which are actually different.
In particular, let us consider the solution
The following point will correspond to this solution
which belongs to the line (4) .
The following point
P,= rp,(A)-cp,(0), C'^')
which belongs to a certain surface (4) making up the surface (3) , will cor-
respond to the solution
which is not actually different from the first.
210
We must now determine whether the surface (.3) Includes layers other
than C^') approaching very close to C^'), i.e., whether there are points /2Q9
on the surface (4') through which other layers of the surface (3) pass
in addition to the surface C^') itself.
Without limiting the conditions of generality, we may assume that
6i = (or we may impose another arbitrary relationship between the g's).
In actuality, the solutions
are not different, and it is sufficient to take one of them into consid-
eration.
We may choose the constant- h p bitrarily, and we may take it in such
a way that, for example,
/.(A)-?.(o),
from which we have
q.e.d.
If we impose this condition g^ = 0, the two surfaces (3) and (4') may
be reduced to curves, and the surface (4') may be reduced to the line (4),
in particular.
We would like to again determine whether another branch of the curve
(3) passes through a point of the line (4) .
For this purpose, let us combine equation gj = with equations (3).
These equations will represent the curve (3) , or a curve of which (3) is
only a part. In the region under consideration, in order that this
curve may not be reduced to the line (4) , it is necessary that the Jaco-
bian ti > 'l'2» •■•> ^-a.* ^1 with respect to Bi, B2> •••> ^n* "^ ' ^^^ ^'^^'- °^
^ly 'f'2. •••> ipn with respect to B2» ^3. •••» ^n» '^ > ^^ ^^^° ^°^ X = 0.
Since nothing distinguishes 6i from other g's, the Jacobians of the
ifj's with respect to t and with respect to n - 1 arbitrary B's must all
vanish. That is, all the determinants included in the matrix of Nos. 38
and 63 must vanish at the same time. By pursuing a line of reasoning
similar to that presented in No. 63, we may see that the equation for S
must have two zero roots.
As a result, two of the characteristic exponents must be multiples of
-^. This is already true for the one of them which is zero. A second
T J -1 fT .inr
exponent must be a multiple of , .
211
If this condition is fulfilled, we shall formulate a system of /210
n + 1 equations including equations (3) and Bi = 0, We shall derive
T and the B's in the form of a series developed in whole and fractional
powers of X •
If the series are real, we shall have periodic solutions of the
second type; if the series are imaginary, this will not be the case.
I shall not continue this discussion any further.
317. Let us now assume that the equations
^ = x. (1)
where time enters explicitly have a uniform integral
in such a way that we have
We saw in No. 64 that in this case the Jacobian of the tjj's with re-
spect to the g's vanishes, and that one of the characteristic exponents
is zero.
The equations
^1 ^ ■■■< i'"-!. F.
with respect to the 3's vanish.
This condition may be written in still another form.
Let us assume that we have solved equation
with respect to x^, and that this solution yields
Let us substitute in place of x^ in Xj^, and let X^^ be the result of
this substitution.
Equations (1) are thus replaced by the following
-^-- = x; (,-.,,. ;,-,). (1-)
These equations (1') will have the following periodic solution
a:,- = 9,(0-
The number of characteristic exponents of this periodic solution,
which is assumed to belong to equations (1'), will be n - 1. Let
ai , a2, ...» Ojj.i be these n - 1 exponents. These will be the same as
those for this periodic solution x^ = i(t), which are assumed to belong
to equations (1), suppressing the n exponents which equal zero.
In order that equations (1) have periodic solutions of the second /212
type in the vicinity of a point on the line (4) , it is necessary and
sufficient that equations (1') have them, i.e., that one of the n - 1
characteristic exponents oij^, 012, ..., ex , is a multiple of -r— at a
213
point on the line (4) .
Thus, the condition, which was presented above, that the Jacobian
of tjji, 11^2. •••» V-l» ^ is ^^^° "'^y ^^ expressed in a completely differ-
ent manner. In order that it may be fulfilled, it is necessary that
two of the exponents be multiples of -^. This is always true for the
one of them which is zero; this must be true for a second exponent.
Let us assume that this condition is fulfilled. From equations (3')
we shall derive the 3's in series which are ordered in whole and frac-
tional powers of X. I shall not extend this discussion, to determine
whether these series are real.
318. Let us now assume that the X±' s do not depend explicitly on
time and that equations (1) have an integral
F = C.
In this case, according to No. 66, two of the characteristic expo-
nents are zero. If the equations have a periodic solution for a system
of values of y and of C, they will still have it for the adjacent values,
so that we shall have a double infinity of periodic solutions
which depend on the two parameters y and C. The period T will not be
constant; it will be a function of y and of C.
Let us then assign a fixed value Cq to C, and let
be the values of x-^ for t = and for t = k (T + t) .
We shall add equation F = Cq, and then an arbitrary relationship be-
tween the e's ~ for example, Bj = ~ to the equations
(3)
Without limiting the conditions of generality, and for the same /213
reason as was given in No. 316, we may assume that 6i = 0.
We shall thus obtain the system
4-v = o, F=^.Co, p, = o. O")
These equations represent a curve. The number of equations equals
n + 2, but the n equations (3) are not different, and may be replaced by
n - 1 of them. This is justified by the same line of reasoning that was
presented in the preceding section. System (3") may thus be reduced to
214
n + 1 equations. The number of variables is n + 2 — i.e..
This curve (3") includes the line
P, = o. W
Let Bj^ = 0, y = yo^^^ point on this line. In order that another
branch of the curve may pass through this point, it is necessary that
the Jacobian of the first terms of equations (3") be zero or — which
amounts to the same thing — that the Jacobian of n - 1 of the ifi's and
of F with respect to B2> ^3: •••> ^n ^"^^^ "^ ^^ zero. Finally, since
nothing distinguishes Bi from the other 3's, it is necessary that the Jaco-
blans of F and of n - 1 arbitrary il^'s with respect to t and to n - 1 ar-
bitrary 6's all be zero.
This condition may be expressed in another way.
Just as in the preceding section, we shall derive the following
from the equation F = Cg
T„ ----- 0(3ri, .T2, .. ., .r„_,)i
and we shall obtain the equations
''^^=Xi (t=.,,a,...,n-.). (1')
tit
According to No. 316, of the n - 1 characteristic exponents, it Is
necessary that one of them be zero and that the other be a multiple of
-r=r [if it is assumed that the periodic solution belongs to equations
(1')]. In other words — which amounts to the same thing — it Is neces-
sary that of the n characteristic exponents [if it Is assumed that the
periodic solution belongs to equations (1)], two be zero, and a third /214
be a multiple of "tzT'
Let us assume that this condition is fulfilled. We shall derive the
6's and the t from (3") In series which are ordered according to whole
or fractional powers of X. I shall still forego a discussion of this
point.
Application to the Equations of Dynamics
319. I would like to discuss the equations of dynamics in greater
detail. However, in order to do this I must first present an important
property of these equations.
215
Let C^ and n^ be the values of x-j^ and y^ for t = 0. Let X^ and Y-j^
be the values of x^ and y^ for t = T. We know that
is an integral invariant. We shall therefore have
with the double integral extending over an arbitrary area A.
This may be written as follows
A(X, dXi - Y, rfX, - ^- dr^i H- T„ dli) -- o,
where the simple integral is extended along the contour of the area A,
i.e., an arbitrary closed contour.
In other words, the expression
S ( X| S]
For T = 0, we have dS = and consequently
S ~ const.
We shall take this constant to be zero so that S will vanish identi-
cally for T = 0. The function S is thus completely determined.
321. Let us determine the maxima and the minima of the function S.
Let us first consider T as a constant. In order that the function S has
a maximum or a minimum, assuming that this function S may be regarded as
a uniform function of the variables Xi + gj and Yj^ + m in the region under
consideration , it is necessary that its derivatives with respect to these
variables are zero — i.e., that we have llld
X, - 5,-, Y,- = 7),.
The corresponding solution is therefore a periodic solution of
period T . and this period T is one of the known quantities of the problem
at hand.
We shall no longer regard T as a known quantity. In order that S
has a maximum or a minimum, it would be necessary that we first have
X; = ?„ Y,= 7;,,
and in addition ^g
However, if X = 5, Y = n, we still have
from which it follows that
^^=.(F_C),
F = C.
217
The corresponding solution will still be a periodic solution of
period T.
However, the period T will no longer be a given quantity. The
energy constant C, which did not enter the preceding case, will be a
given quantity.
The two methods for determining the maxima of S are related to the
two methods of interpreting the principle of least action, that of Hamil-
ton and that of Maupertuis. This will be clear to the reader after the
following chapter has been read.
322. The definition of the function S may also be modified in the
following way.
In a large number of applications, F is a periodic function of
period 2n with respect to the y^^'s. In this case, a solution may be
assumed to be periodic when ^± = ^±, and when Y^ - m is a multiple of
2tt.
It is then apparent that if we set
where m^ , m2, . . . , 1% are arbitrary whole numbers, the expression dS
will still be an exact differential.
We shall thus obtain /217
dS
rfT
We shall set
For T = 0, we have
dS ==S4/;i,TCrff/.
We shall set c , v ^
which concludes the determination of the function S.
Assuming that T is a given quantity, the maxima and minima of S will
be obtained by setting its derivatives equal to zero, which yields
218
'^^ aF-sFcx-S) )^| +(Y-r, — 2«ir)^'^^l+ arbitrary function of T.
X,- = I/, Yf = T^i + 2;n,Tr.
The corresponding solution is still a periodic solution, since
Y^-Tii is a multiple of 2tt. The period T is given.
If T is not given, it Is first necessary that
and, in addition.
>'', = ?i, Y,— r;,-{- 2/n/Tt
rfS
from which we have
dT = "•
Fr=C.
323. It is now necessary that we learn to distinguish between the
real maxima and the real minima of S. Up to this point, we have only de-
termined the condition for which the first derivatives of S are zero, but
it is known that this condition is not sufficient for providing a maximum.
It is still necessary that the second derivatives satisfy certain inequali-
ties.
Let us first assume that the conditions presented in No. 319 hold,
and let us regard T as given.
Let
^/ = ?-('). r<--?HO
be a periodic solution of period T, so that J218
tp,(o) = 9;.f, o;,-, e;.,
will be imaginary and conjugate of
0<-.6 0/,.n QI.;, or.,-.
Let us now assume that the x'. 'g and the y^^ 's are real. In order to
calculate the constants A, B, C, D, we shall have 2n equations which we
shall obtain, for example, by setting the following in the equation for
x'.
1:^0, l = T, t=-.-2T, ..., / = (2n — i)T.
These 2n equations are linear with respect to the 2n unknowns A, B, C,
D. The second terms are real, and the coefficients are real or imaginary
and conjugate pairwise.
When we change \^ - 1 into -y-l:
1. Aj^ and Bj^ do not change when aj^ is real;
2. Aj^ and B, interchange when oj^ is purely imaginary;
3. Aj^ and Bj^ change into Aj and Bj when a]^ is complex and imaginary
and conjugate of cxj .
Therefore:
1. Aj^ and Bj^ are real when ajj^ is real;
2. Aj^ and B^^ are imaginary and conjugate when aj^ is purely imaginary;
3. Ajj^ and A., Bj^ and B. are Imaginary and conjugate when a,^ is /222
complex, and imaginary and conjugate of a ^ •
Finally, C and D are real.
These conditions are sufficient for x?j^ and y'j to be real.
Let us assign values satisfying these conditions to the constants
A{j^, B}^, C, D, as well as to the constants Aj^, Bj^^, C' , D' . Then the
second term of (2) must be real, and in order that it may be real the
following is necessary:
1. That Mj^ is real if ay^ is real;
223
2. That Mn is purely Imaginary if ay^ is purely imaginary;
3. That tiy^ and M. are imaginary and conjugate if a^ and a are
complex, and imaginary and conjugate.
Form (3) contains a term
and does not contain another term depending on Aj^ or \.
If the exponent a^ is real, the presence of a term containing Aj^B^
is sufficient for providing that the quadratic form (3) can be defined.
Therefore, if only one of the exponents aj^ is real, the function S
cannot have either a maximum or a minimvmi.
Let us now assume that two exponents aj^ and a^ are complex, and
imaginary and conjugate.
Let us cancel all the constants except for
Ai, B<-, Ay, Bj,
aid the form (3) may be reduced to
These two terms are imaginary and conjugate, so that form (3) is real.
Let us assume that Aj^ does not change, and that B^ changes sign.
A., which is imaginary and conjugate of Aj^., will change no longer, and
B., which is imaginary and conjugate of B^, will change into -Bj .
Therefore, form (3) will change sign; therefore, it cannot be defined.
Therefore, if only one of the exponents aj^ is complex, the function
S cannot have either a maximum or a minimum.
Let us now assume that ay^ is purely imaginary. Then A^ and Bj^ are /223
imaginary and conjugate, and the product \By^ is the sum of two squares.
In order that S have a maximum, it is necessary and sufficient
that all of the quantities
^sin^^, -NT
/ - ■ I y/— I
224
be negative. In order that S have a mlnimian, it is necessary and suffi-
cient that all these quantities be positive.
It should be pointed out that all these quantities are real, because
^ and ^ are real.
325. How may these results be modified if it is assumed that the
energy constant is one of the given quantities of this problem? We then
have identically
/dP , (IF ,\
dF dF
where we assume that in — and -j— , x^ and y^^ have been replaced by the
periodic functions ^^{.t) and ijj^Ct).
The constant value of the function F must be the same for the periodic
solution
and for the infinitely close solution
xi --: y,(0 + ^'h yi --= iM+y'i-
This relationship is a linear equation between the constants
Ai, Bi, C, D
and the coefficients must be independent of t.
It follows from this that Aj^ and B^ must not be included in the re-
lationship, since these constants are always multiplied by e^^^k and
since this exponential cannot vanish.
In addition, C is no longer included, since the solution
a^i = ?/(«j-HC ^^y, y,-. = 7,(0-1 „_,,
2r, o, ..., o,
o, 2-, ..., 0,
* • • . . . ,
o, o, ..., ■y.r.,
will play the role of periods in this line of reasoning.
There would be an exception, if these periods were not different —
i.e., if one of the quantities oj were connnensurable with Zir, or, more
generally, if there were a linear combination of the z's which had only
one single period — i.e., if there were a relationship having the form
^iWi+ 6,(0,-)-. . .4- 6„_,w„^i-t- 2-ii6„ = o, ^2)
where the b's are whole numbers.
Let us disregard the case of this exception. The quantities (1)
will equal
—^=z 5111 Zl.
We may choose the whole nvraiber m in such a way that these quantities repre-
sent a combination having a given sign — i.e., that there are numbers
zj^ which satisfy inequalities having the form
«1<-Il< rtl 1- 'T, fll < Jj < ai -t- Tt, ,.., fl„_,< 2„.-i< rt„_,^-1r, y-o\
where the aj^'s equal or tt.
This results directly from the statements which we have just /228
made above .
Let us move on to the case in which we have a relationship of the
form (2) . We may always assume that the whole numbers b are primes among
themselves. In this case, the expression
has only the period 2Tr.
In order that there may be no numbers z, satisfying the inequalities
(3) , it is necessary and sufficient that the difference between the larg-
est value and the smallest value which expression (4) takes — when all
values which are compatible with the inequalities (3) are assigned to be
z^'s — is smaller than 27r, i.e., smaller than a period of this expression
(4).
This difference is obviously as follows
229
and we must therefore have
|i,l-t-lfcil-t-.---t-l*'.->l-^- (^5)
The inequality can only hold if all of the b's are zero, except for
one of them which must equal +1.
In this case uiy, must equal a multiple of 2t7. This means that a^
must be zero, since a^^ is only determined up to a multiple of ---^
We have excluded the case in which one of the aj^-'s is zero.
The equation can only be valid if all the b's are zero, except for
two of them which must equal +1.
Then the sum of the difference between two of the uj^-'s will be a
multiple of 2tt. If we note that the a^^'s are only determined up to a
multiple of HL/Z^, we may express this result in another way.
Two of the characteristic exponents will be equal.
This is the only exception which still exists, and it may be readily
excluded.
329. Let us now assume that the equations of dynamics under consid-
eration depend on an arbitrary parameter y, just as is the case for the /229
three-body problem, as we know.
When we vary y continuously, the periodic solution
will also vary continuously, as we may determine from the discussion
in Chapter III.
The quantities Mu will also vary continuously, but ~ as was ex-
plained in No. 323 ~ they can never vanish. Therefore, they will
always retain the same sign , and It is their sign alone in which we are
interested.
The energy constant will be regarded as one of the given quantities
of the problem at hand, but this given quantity may depend on y, and we
230
shall choose it in such a way that the period T of the periodic solu-
tion remains constant.
The exponents a^^ will also vary continuously when we vary y con-
tinuously. Let us clarify to a certain extent the manner in which this
variation should be handled in the case of the three-body problem. For
y = 0, all the exponents are zero. However, as soon as y ceases to be
zero, the exponents cease to be zero also. One of these exponents can
only vanish, or become equal to a multiple of ^^^^^ , or become equal
to another characteristic exponent for certain special values of y.
330. Let us consider a periodic solution of period T, such that
all the exponents aj^ are purely imaginary. This is what we designated
above by a stable solution. In Chapters III and IV, we proved the
existence of these solutions.
Let us consider one of the exponents, cxi, for example. When y varies
continuously, T^fj' — which is real — will become commensurable with
27T . ^. .
— an infinity of times. Let us assign a value yQ to y, such that
where k and p are the prime whole numbers among themselves. In add- /230
ft .
ition, this value does not correspond to a maximum or a minimum of
At a later point, in No. 334. we shall see why I have placed 2kTT
in the numerator, and not kir .
In any interval, no matter how small it may be, there is an infinite
number of similar values.
If m is an arbitrary whole number, for this value yg the expression
/- . / "-7
is zero. In addition, since yg does not correspond to a maximum or a
minimum of -y-'^-' this expression will change sign when y passes from
yg — e to yg + e.
For example, let us assume that it changes from being negative to
being positive.
231
Pursuing the line of reasoning presented in No. 328, we will find
that we may choose the whole number m in such a way that the expressions
— -= tin ^r-^- (A— 2, 3, . . . , n — i)
y/ - I v'-^ I
have all possible combinations of signs, and that they are all negative.
Under this assumption, for m =- Mq - c, our function S^^^p will have
a maximum, since all our expressions will be negative. However, for
^jj = ^Q + e^ our periodic solution will no longer correspond to a maximum
of S , since one of these expressions will have become positive.
Theorems Considering the Maxima
331. In order to pursue this subject further, it is necessary to
illustrate one property of the maxima. Let V be a function of the three
variables xi , X2 and z, which may be developed in increasing powers of
these three variables. I shall assume the following:
, , . dV dV
1. For XI = X2 = 0, V vanishes as well as its derivatives ■^, ^,
no matter what z may be;
2. For xi = X2 = 0, V has a maximum for z > and a minimum for /231
z < 0.
It may be stated that the equations
dxi dxx
have other real solutions in addition to the solution
Let us develop V in powers of z, and let
The functions Vq , Vi , V2 , ... may themselves be developed in powers of xi
and of X2. However, these expansions will contain neither terms of
degree nor terms of degree 1, because — no matter what z may be ~ we
must have
V - i— - -- -
dx^ dxf
for xi = X2 = 0.
In addition, Vq does not contain terms of the second degree either.
232
Without the second degree terms, it is impossible to pass from the case
of the maximum to the case of the minimum, when going from z > to
z < 0.
Conversely, Vi will contain first degree terms, at least we shall
assume this is the case. Let us then consider the equations
dxi dxi clxt . .
o = -J i- z -.— -t- j' --, H . . .
rfr, dX} clXi
which must be solved.
Let Uq and Ui be the lowest degree terms of Vq and of Vi . According
to the statements which we have discussed, Ui is of the second degree,
and Uq is of the degree p ~ with p being larger than 2. Let us set
{p~7.)iL^.i; x,=y,i, x,^-r,t, \ = ^ytp■, 3==±tp-'
W may be developed in powers of t. Let us set
^v== w,-r- AV,i-i-Wj-i-....
We obviously have /£3£
U\ = Uit~P and Uq = Uot~P are two homogeneous polynomials in yi and y2 —
one of degree 2 and the other of degree p. I shall employ the sign + or
-, depending on how I have set z =+tP"2. The expression
dV dV, dV dV,
dXf dxx dxi dxx
will also be developed in powers of t when xi and X2 are replaced by
yit and yat. It will include a certain power of t as a factor. Let us
divide by this factor, and let H be the quotient. This quotient developed
in powers of t may be written
11 =no+HIi+<'H, + ...;
Hq will be the first of the expressions
dWj. dU'i _ dyVt dU\
dy, dyi dyt dyi
which will not vanish.
The equations ^y rfV
dXi dXi
233
may be replaced by the following equations
II = o -,— = o.
I shall prove that we may derive the y's from these equations in the
form of series which are ordered in fractional and whole powers of t,
which vanish with t and which have real coefficients.
In order to do this, according to statements presented in Nos. 32
and 33, it is sufficient to establish the fact that for t = 0, these
equations have a real solution of odd order .
For t = 0, these equations may be reduced to
llo = o, -^=c,
or /233
rfw< dv\ _ d\yj, cnj\_ __
and
rfU', d\}'
Equation (2) indicates that Wi^^ has a maximum or a minimum, if we
assume that y\ and y2 are related by the relationship U'j = const.
For the present, if we assume that yi and y2 are the coordinates of
a point in a plane, the relationship U', = const will represent an ellipse,
because the quadratic form Ui (and, consequently, the form up must be
defined in order that V may have a maximum or a minimum. Due to the
fact that an ellipse is a closed curve, the function W2 must have at
least a maximum and a minimum when the point y^, y2 describes this
closed curve.
Therefore, whatever the constant value may be which is assigned to
Uj^, equation (2) will have at least two roots, and two roots of odd order ,
because we have seen in No. 34 that a maximum or a minimum always corres-
ponds to a root of odd order. At this point, where we have no more than
one independent variable, the theorem presented in No. 34 is almost self-
evident. Under this assumption, we may distinguish between two cases:
First case . Uq is not a power of \^y In this case, we do not have
identically ^^o d\]\ d\\\ d\}\
d^i dyi dyi dyi
= o.
234
We shall therefore have W]^ = Wq , and
dU' dV\ dU' dU'.
Equation Hg = is then homogeneous in yj and yi . No matter what the
constant value is which is assigned to U', , it will provide us with the
yi
same values for the ratio — .
72
We may derive — from equation (2) and, according to the preceding
statements, we shall obtain at least two solutions of odd order. /234
vi °'i
Let -i-i- = — be one of these solutions. Let us set vi = aiu, yo =
= a2U and let us substitute in equation (3) . We shall have
and equation (3) may be reduced to
If p - 2 is odd, this equation will give us a real value for u.
If p - 2 is even, we may distinguish between two cases.
If A and B have the same sign, we shall take the lower sign
A «/>-'— 15 =0.
If A and B have opposite signs, we shall take the upper sign
Ait/'-s ;-B :;-- o,
and we shall have two real values for u.
In every case, these real solutions are simple.
Thus, equations (2) and (3) will always h.ave solutions of odd order.
Second case . We have
u; = A(u;)f.
We shall begin by solving equation (3), which may be written as
follows
p ''-1
:^A(U',)' ±1 = 0.
2
235
This equation provides us with the value of U'^ . This value is
real and simple, but this is not sufficient because U*^ is a negative
definite form. In order that the solution may be suitable, it is
necessary that the value found for U'^ be negative; as a consequence,
we shall choose the sign +.
The value of U^ having thus been determined, we may assign this
constant value to U'^ , and in order to solve equation (3) we need only /235
determine the maxima and minima of Wj^. As we have seen, we shall derive
at least two solutions of odd order.
We have therefore established the fact that equations (2) and (3)
always have real solutions of odd order. The theorem presented at the
beginning of this section has thus been proven.
332. Now let V be a function of n + 1 variables
T], 2-2J ••■> ^n and z.
I shall assume the following:
1. V may be developed in powers of x and of z;
2. For
a^i m 2*3 = . . . = Xfi =^ o,
we have the following, no matter what z may be
rfV _ £/ V _ _ ^v _
dx^ Jj-, ' ' ' d.T„
3. Let us consider the group of terms of V which are second degree
terms with respect to the x's. They represent a quadratic form which may
be equated to the sum of n squares having positive or negative coefficients.
When z changes from positive to negative, I shall assume that two
of these n coefficients change from positive to negative, and that the
n - 2 other coefficients do not vanish.
Under these conditions, it may be stated that the equations
^ _ JV _ _ ^ _ (1)
dXi dxt ' ' ' dx„
have real solutions which differ from
.Ti — rj = . . . = ^n " o.
236
Let us develop V in powers of z and let us set
Let Uq and U^ be the group of second degree terms of Vq and Vj .
The group Uj is a quadratic form which may be decomposed into a sum
of n - 2 squares, because we know that, for z = 0, two of the coefficients
which were in question above vanish. /236
Therefore, if we consider the discriminant of Uq, i.e., the func-
tional determinant of
(/j-, ' c/.r, ' ' d.fn
with respect to
■^'ii ^i, ■■-, ■i"/i,
this determinant vanishes, as well as all of its minors of the first
order. However, all of the second-order minors do not vanish, unless a
third coefficient is zero, which we have not assumed.
We may also assume that a linear change in the variables has been
performed, so that Uq is restored to the form
Consequently, the functional determinant of
(Wo 0, V — regarded as a function of x^ and of X2, —
has a maximum when these two variables are zero.
In order to illustrate this, we must try to find the second-degree
terms with respect to x^ and X2 in V. Let
238
Wo 4- 3W,-h-'-W,+...
be these terms. In order to obtain
W„-i- j\V,
which are the only ones which interest me, I shall take the two /238
terms
Uo-l-3U,,
and I shall neglect the other terms of V which cannot influence Wq + zWi .
I may derive the following from equations (2)
in the form of series ordered in powers of xx and X2. In these series,
I shall only retain the terms which are of degree 1 with respect to x^
and X2, and of degree with respect to z. The other terms may be neg-
lected, because they do not influence
Wo+aW,.
Equations (2) may then be reduced to
1 A i, .Ti + z - - — -- o,
. rfU,
i.\„Xn'^ z - — — o.
dXn
If we substitute the values thus obtained in place of X3, xt^, ...,
Xjj, in Uq, we shall find that Uq is divisible by z^. With respect to
Uj, it may be reduced to
where Uj is none other than the quantity which Uj becomes when we cancel
X3, xi^, ..., Xjj, and where u] and U^ are two other quadratic forms with
respect to the x's. We shall therefore have
and
Uo-f- ;:U, == 5 U}H- ;:2(U3 -H Ul)+ ;:3 UJ.
239
In order to calculate Wq + zWi , I may neglect the last two terms
which may be divided by z^ and z^, and I shall simply have
I shall demonstrate the fact that V has a maximum for xi = X2 = /239
and for z which is positive and which is very small. It is sufficient
to illustrate this for Wq + zWi, i.e., for zU^ .
Finally, we must prove that uj is a negative definite form.
For this purpose, we shall write the quadratic form Ui as follows
u; is a sum of two squares having coefficients whose sign I shall not
predict. U" depends only on the n - 2 variables
This is always possible, according to the general properties of quadratic
forms.
Let lis consider the form
where z is assumed to be positive and very small. The form U^ + zU'^' ,
which depends only on the n - 2 variables X3, x^, ..., x^^, may be equated
to a sum of n - 2 squares having coefficients whose signs must be the
same as those for A3, A^, . . . , A^, since — due to the fact that z is
very small — this form differs very little from Uq. Therefore, they do
not change sign when z makes a transition from positi ve to negative.
According to our hypotheses, when z makes the transition from posi-
tive to negative, n - 2 of our coefficients do not vanish, and, on the con-
trary, two coefficients make the transition from negative to positive.
These last two coefficients can only be the coefficients of Ui.
Therefore, V[ is the sum of two squares having negative co efficients.
In order to have u", it is necessary to set the following in U'^
iCj = T^ = . , . =^ 37/1 O,
Then U" vanishes, and U^ may be reduced to U'^
Therefore, U^ is a negative definite form.
q.e.d.
240
Therefore, V, regarded as a function of x^ and X2, is maximtan for
z which is positive and is very small, and for x^ = X2 = 0. /24Q
One will find in the same way — or rather one will find at the
same time — that V is minimum for z which is negative and very small,
and for x^ = X2 = 0.
As I have stated, we have thus returned to the conditions stipu-
lated in the preceding section, and it may be assumed that the theorem
presented at the beginning of this section has been substantiated.
Existence of Solutions of the Second Type
333. Let us return to the hypotheses given in No. 330. We have
defined the function S^jj- , which depends on y, of the 2n variables
The Sj^'s and the ri-j's are the values of x^ and y-j^ for t = 0. The
Xj^'s and the Y^'s are the values of Xj^ and y^ for t = mpT.
We sould like to study the solutions of the equations
According to Nos. 321 and 322, these solutions correspond to periodic
solutions of period mpT. We already know one of them, since a periodic
solution of period T is at the same time periodic having the period mpT.
I propose to show that there are others in addition.
First, however, I would like to illustrate the method which may be
employed to regard S as being dependent only on p and on the 2n - 1
variables
( X| + ^,, Xj-H^j, ..., X„-i + ^„_,, , .
( Yi-t-T,,, Yj-T-T,,, ..., Y„_,4-T)„_,, Y„ + i]n.
For this purpose, we shall assume that
Let us now consider the equations
dS„
J(X(H-$,-) OCii-i-r,i)
(1')
We shall employ the d's to represent the derivatives of S which is
assumed to be a function of the variables (a), and shall employ the 3's
241
to represent the derivatives of this same function S which is assumed /241
to be a function of the variables (B) .
I plan to show that equations (1) and (1') are equal.
Section No. 322 has provided us with the following
dS = Z[( \i - I) diY, + r,i) - ( Y; - T„- - 2 m,7!}d( X, -4- r„-)].
Equations (1) may therefore be written
— ( Y, — -/,,■ — ■> »),-) -= X, — ^,- = o
(' ^ 'i ' ")>
and equations (1') may be written as follows
— (Y/ — T,, — -imtr.) -. \i — u ^o
{i ^ I, 1, ..., n ~i),
X„ -f, — o.
In view of the energy eqiiation, we have also
According to equations (1'), all of the X^'s equal the C^'s, and all
of the Y.'s (except one) equal n^^ + 2m-j^ir. The preceding identity may
therefore be written as follows. For purposes of abbreviation, I shall
write
F(;i. ;!> ■•■. ?«; T,i-HQ7n,T7, rij-t-iOTjTT, ..., T,„_,-T--2m„^|-;t, Y„) = F(Y„).
My identity may therefore be written in the following form
F[r,„ u 9. m„ 77 + ( Y„ — T), — 2 /«„-)] - 1? (t], ■;- ini^Ti) -^ o,
or, in view of the theorem of finite increases
(Y„ — T,„ — a77i„T:)F'[Tj„-+-2m„^ -i-0(Y„ — T,„ — jm„-n:)] = o, (2)
where 6 is included between and 1, and where F' is the derivative of
F with respect to Y^^.
Let i. and n? be the values of E,^ and n^ which correspond to the
periodic solution of period T. The region under consideration only in-
cludes the immediate vicinity of the point \i = \iq, E,^ = g_^, t], = n^.
Therefore, 1,^ and X. will never deviate greatly from 5?, and n^ or
Y. - 2m^TT will never deviate greatly from n^. Therefore, the second
factor F' of relationship (2) will never deviate greatly from its value
242
^ _
tor ^^ - K -^t '^1 ~ ^i' ^ ^^ general this value will not be zero.
Therefore, the first factor of relationship (2) must vanish, and /242
we have
Y„— T,,— im„TX =r o.
In other words, equations (1') entail equations (1). We may
therefore regard S as a function of the variables (B). When it is
a maximum, considered as a function of the variables (B) , it will also
be a maximum as a function of the variables (a) .
I have employed 5? and n? to designate the values of C^ and of n^
which correspond to the periodic solution of period T. The corresponding
values of X^^ + ^^ and Y^ + m will be 2^^ and 2n9 + 2m£mpTr (if the periodic
solution of period T changes y^ into y^ + 2m^7T, in conformance with the
hypotheses formulated in No. 322). Let Sq be the corresponding value of
Sjjj . Let us set
and let us consider V as a function of y' , of the 5' 's, and of the n' 's .
The function V will be governed by the same conditions as the function V
of the preceding section.
No matter what y' may be, V and its first derivatives with respect
to the C 's and to the n' 's will vanish when
?;-=T/i---o.
If we consider the group of second degree terms of V with respect
to the 5' 's and the n' 's , and if we regard it as one quadratic form
which is decomposed into a svrai of square terms, it may be seen that two
of these coefficients of these square terms both make a transition from
negative to positive, or both make a transition from positive to negative,
when y changes sign. The other coefficients do not vanish.
The expression
— _ —sini — .
changes sign, and the other expressions
: sini
/ITT /:
243
do not vanish. The coefficient which I have designated as D in No. 323
no longer vanishes, and there is not another one because we have only /243
2n - 1 variables , the variables (6) .
The conditions presented in the preceding section therefore hold,
and we may state that the equations
rfV d\
d\i drii
have other real solutions in addition to E.'^ = n'^ = or, which means
the same thing, equations
i'^mp ^ ^ ( f^mp _ g (1)
have other real solutions other than those corresponding to the periodic
solution of period T.
The maxima of the function Sjjjp, or more generally the solutions of
equations (1), correspond to periodic solutions of period mpT.
We must therefore conclude that our differential equations have
periodic solutions of period mpT, which differ from the solution of
period T, which is identical to that for y = yo. and which differ only
slightly for y close to yg.
If attention is drawn to the preceding line of reasoning, we shall
find that the periodic solution of period T need not correspond to a
maximum of S^j-.
We shall therefore set m = 1.
It is not necessary that the solution of period T be stable. It is
sufficient that one of the characteristic exponents ai equals
for y = yo.
We therefore obtain the following result.
If the equations of dynamics have a periodic solution of period T,
such that one of the characteristic exponents is close to
they will also have periodic solutions of period pT which differ very /244
244
little from the solution of period T, and which are identical to the
latter when the characteristic exponent equals
These are solutions of the second type.
Remarks
334. This entire line of reasoning assumes that S^p is a uniform
function of X^ + g^ , Y^ + ri-j^ . Under this condition alone may it be
stated that all the maxima of S correspond to a periodic solution
(see No. 321). This fact cannot be stressed enough. It is an obstacle
which will be encountered frequently when we wish to derive the results
of the theorem presented in No. 321.
Let us determine whether S is a uniform function of these variables.
We may assume that m = 1, which we have just illustrated. In addition,
Sp is clearly a uniform function of the 5^'s and the Hi's. It will also
be a uniform function of the X^ + ^^'s
and the Y^ +
n^-s.
provided that
the functional determinant of the X^ + 5^'s and the Y^ +
ri-j^'s with
respect to the C^'s and the n^'s does not vanish in the region under con-
sideration. Due to the fact that this region may be reduced to the imme-
diate vicinity of the values
J'o,
5.= $?,
n?.
it will be sufficient that the functional determinant is not zero
at this point. This functional determinant may be written as follows
(assuming that n = 2 , to formulate our ideas more clearly)
^^' u ,
'l^L
dXi
dX,
/f, -"- '
(Ir^'t
dU
dM
dY,
'dU
dY,
d'ii
d'n
dX,
dU
dX,
dn.
dY,
dY,
dY,
dY,
dU
dra
di.
dn.
It must therefore be verified that the equation in S
/245
245
d\,
dhi
dX,
'du
dX,
dr„
4.
dn 1
d\\
'dU
dr^
d\i
dni
d\r ^
-df.-^
dX,
dr,.
d\.
,?Y,
d\.
<^Y,
'du
dfj,
d-t
dr„
does not have a root which is equal to -1.
According to the statements presented in No. 60, the roots of this
equation equal
where the a's are characteristic exponents. We must therefore verify
the fact that we do not have
/'I
where k is an integer number. By hypothesis, the exponent a^ equals
2 f>~ v/-7
where k is an integer number, and the other exponents are not commensur-
able with!L./:il. , in general.
T
The difficulty with which we are concerned will not therefore
occur.
In order to avoid this, in No. 330. I assumed that
,'iTi/^7 (.^ integer number)
and not _
/,Trv/£_|_
''' ■ pT (k integer number)
Special Cases
335. Let us say a few words about the simplest cases, and let us
assume only two degrees of freedom.
Let us assume that the form which is similar to that which I have
246
designated as Uq , In the analysis of No. 331, is homogeneous of the /246
third degree only in xi and X2 .
The equation
dxl Jzj clxl Z^ ^- " (1)
always has real r 0, a maximum for p = and two minima.
For z < 0, a minimum for p = and two maxima.
Employing the English term, I shall use the word minima to desig-
nate a point for which the first derivatives vanish, and where there
is neither a maximum or a minimim.
The same will hold true for the function V, since — if z is
very small — the terms Uq + zUi alone will have an influence.
Therefore, no matter what z may be, the differential equations will
have :
247
^^
A solution of period T, of the first type, which is stable;
A solution of period pT. of the second type, which is stable for
z < and unstable for z > 0.
Let us now assume that equation (1) has three real roots.
The function f (<(.) will have three maxima and three minima which are
equal pairwise and have opposite signs.
In this case Uq + zUi , and consequently, V have:
For z > 0, a maximum for p =0, and six minima;
For z < 0, a minimum for p = 0, six maxima.
No matter what z may be, the differential equations will therefore
have:
A solution of period T, of the first type, which is stable;
Three solutions of period pT, of the second type. We shall see
below that, from a certain point of view, none of these solutions are
different.
Let us proceed to a case which is a little more complicated, and
let us assume that Uq is of the fourth degree.
In this case, equation (1) is of the fourth degree, and, since it
always has at least two real roots according to No. 331, it will have
two or four. We then no longer have
but rather
/()-/('f-H^)-
Let us first assume that there are only two real roots.
The function f ((j>) will then have a maximum and a minimum when (t>
varies from to tt, as well as when ^ varies from u to 2tt.
A distinction may be drawn between two cases, depending on the signs
of this maximum and this minimum.
First case . The maximum and the minimum are positive.
The functions Uq + zUi and V have:
248
For z > 0, a maximinn for p = 0, two minima and two maxima.
For z < 0, a minimum for p = 0.
In addition to the solution of the first type which always exists,
the differential equations have two solutions of the second type for /248
z > 0, and do not have any for z < 0. Of these two solutions, one is
stable and one is iinstable.
Second case . The maximum is positive, and the minimum is negative.
The constants Ug + zU^ and V have:
For z > 0, a maximum for p = 0, two minima;
For z < 0, a minimimia for p = 0, two minima.
The differential equations always have an unstable solution of the
second type, in addition to the solution of the first type which is
stable.
Third case . The maximtm itself is negative.
The differential equations then have:
For z > 0, a solution of the first type which is stable;
For z < 0, a solution of the first type which is stable, and two
solutions of the second type of which one is stable and one is unstable.
We must now examine the case in which equation (1) has four real
roots.
The equations then have:
For z > 0, a solution of the first type which is stable, h solutions
of the second type which are unstable, and k solutions of the second type
which are stable;
For z < 0, a solution of the first type which is stable, 2 - h solu-
tions of the second type which are stable, and 2 - k solutions of the
second type which are unstable.
The integer numbers h and k may take the following values, depending
upon the signs of the maxima and the minima of f ((J>) :
A = ^ r= 2 ; A = 2, /: — i; h = ■>., k = o\ A =- i , k —o;
h = k =o; A = A- = 1 .
249
\-/
CHAPTER XXIX
DIFFERENT FORMS OF THE PRINCIPLE OF LEAST ACTION
336. Let
Ji, 72. ■■•. 7"
be a double series of variables, and let F be an arbitrary function of /249
these variables. Let us consider the integral
,=jr"(-,M..„5),„.
The variation of this integral may be written as follows.
In order that this variation may vanish, it is necessary that we
have
jxi _ ,/v_ -^— -^^' -^^~iJ' ■■■' dx, dt J'
(7)
In the place of the y^'s, let us substitute their values (7) in J
and in the equation /253
F =^0.
dx, dx]j
^1
We shall derive -tT" as a function of the xj^^'s and the -r — 's from this
253
equation. We shall then substitute this value of -^ in expressions
(7) and in J. This last integral will take the following form
/^/'£:'^--/"'^^"
dx]^
where $ is a function of the xj^'s and of the derivatives ■^—. This
integral, which is thus written in a form independent of time, is still
minimum. This is the principle of least action in its Maupertuis form.
If h were not zero, we would only have to change F into F - h.
337. Let lis first examine the most important particular case.
Let us assume that we have
F =^ T — U,
where T is homogeneous of the second degree with respect to the variables
y., while U is independent of these variables.
We then have
Zj^^^-^^-T, II = T-i-U.
According to the principle of Hamilton, the integral
r '(th-u)(/«
must be minimum.
Let us determine what the principle of Maupertuis becomes. The
energy equation may be written
T U = h.
The Maupertuis action then has the following expression
^(T4-U + /t)(/^
The equations ^^^^ ^ £{F = i^
di d/i dyi
have their second terms which are linear and homogeneous with respect to
254
the y^ s. Therefore, T is homogeneous of the second degree with /254
respect to the "^ s . Let di"^ represent that which T becomes when
dx^
-J— is replaced by dx^ ; we shall have
T - - -
and di will be a form which is linear and homogeneous with respect to
the n differentials dx^. We may deduce the following from this
di= ''' '^^
The Maupertuis action will then have the following expression
■>. fdz/uT-Jt.
338. For purposes of brevity, in order to be able to study other
particular cases, let us set
, d.T(
and let us derive the y-j^'s of the equations
,_ dP
so as to take the x^'s and the x^ 's for new variables. Let us employ
the ordinary d's to designate the derivatives taken with respect to the
Xj^'s and to the y^'s, and let us employ round 8's to designate the deriva-
tives taken with respect to the Xj^'s and the x'^ 's.
We may readily obtain the well-known relationships
drj- O.Tj dxt
and we will see that equations (1) are equivalent to the Lagrange equa-
tions
dt dx'i Oxi
/255
Under this assumption, let us examine the case in which H has the
255
following form
H = H,-HH,-t-lI,,
where Hq , H^ , H2 are homogeneous, of degree 0, 1, 2, respectively,
with respect to the variables x'^.
We then have
£ici:r-^ =2H,-t-n,,
F = n,— Ho
and the _ m^ dH,
^' " dx'i "^ d^
are linear functions, but they are not homogeneous with respect to the
■k\ 's.
The Hamiltonian action retains the same form
Al dt.
Let vis determine what the Maupertuis action becomes.
Let h be the energy constant. The Maupertuis action will have the
following expression
r{H-i-h)dt
but it must be written in the form which is independent of time.
For this purpose, let us set
and ,
IT - "
H2 is nothing other than energy, and dx^ is that which this energy be-
comes when x^ is replaced by dx^. In the same way, do is that which
Hi becomes when x'^ is replaced by dx^. It is therefore a form which is
linear and homogeneous with respect to the differentials dx^.
If we take the energy equation into account
H,= Ho+A,
256
from which we have
the Maupertuis action will become /256
The Maupertuis principle may therefore be applied to the case in
which we are interested, as well as to that of absolute motion. However,
there is one essential difference from the point of view of the following
statements .
In all the problems which will be encountered, the energy T or H2
is essentially positive; it is a quadratic, positive definite form. In
the case of absolute motion (No. 337), the action
is essentially positive. It does not change when the limits are inter-
changed. On the contrary, in actuality, the action is composed of two
terms. The first
f'. d: v/Ij;
h
is always positive, and does not change when the limits are interchanged.
The second J da changes sign when the limits are interchanged, and
it may therefore be positive or negative.
If we also note that in certain cases, the first term vanishes with-
out the second term vanishing, we will find that the action is not always
positive. This fact will cause a great deal of difficulty later on.
339. In order to show how the preceding considerations may be
applied to relative motion, let us first consider the absolute motion
of a system. Therefore, let
H = T -H U
and let us assume that the position of this system is defined by n + 1
variables
X|, Tt, ..., X„, O),
where xi , X2, ..., x^ are sufficient to find the relative position of
different points of the system, and o) is the orientation of the system
in space.
257
If the system is isolated, U will depend only on x^, X2, ..., x^^.
T will be a form which is quadratic and homogeneous with respect to /257
x' , x' ..., x' , cj' whose coefficients depend only on Xj^ , X2, ..., x^^.
We will then have the equation
where p is a constant. This is the area integral.
Under this assumption, let J be the Hamiltonian action
J= f Udt;
We shall have the following, if the equations of motion are satisfied
5J -_^\y ^?,r+~ ou)l'"''
The action will be minimum (or rather its first variation will be
zero) if the initial and final values of the x^'s and of u are assumed
to be given — i.e., if 6x^ = 6a) = for t = tg and for t = tj.
Let us now assume that the initial and final values of these x^'s
are given, but not those of w. We shall have
Then let
and
r= Tll'dt,
and we shall obviously have
We may derive to', which is a linear, nonhomogeneous function of
the x'.'s, from the equation x", = P- It may also be seen that H is a
1 dw
quadratic function which is not homogeneous with respect to the x^ 's.
H' therefore has the form Hq + H^ + H2 which was studied in No. 338.
258
The integral J' will thus be minimuni, even though the initial and
final values of cj are not assumed to be given.
We have /258
J' - J — jD('Ji — i)o))
where wg and o)]^ are the values of co for t = tg and t = tj.
340. Let us now assume that we have a system referred to moving
axes and subjected to forces which depend only on the relative situa-
tion of the system with respect to the moving axes. In addition, let
us assume that the axes rotate uniformly with a constant angular velocity
This problem may be directly related to the preceding one. We need
only assign a very large moment of inertia to the moving axes, in such
a way that its angular velocity remains constant.
For the absolute motion, we then have
II .^ T -4- U := T, -*- T, + U.
The function of the forces U depends only on the variables x-j^ which
define the position of the system with respect to the moving axes. Ti,
which is the energy of the system, depends on the Xj^'s, and is a quad-
ratic form with respect to the x^'s and to u' . T2, which is the energy
of the moving axes, equals
2
and the moment of inertia I is very large.
We then have
and
H'= H-j3u' = (T,-T-T,-i-U)- ^ to'-Iu'«"
°^ dT, Ito'«
H'= T,-hU-5^u>'-:^^.
atij 2
However,
1 r "' 1
259
K^
dTi
Since I and p are very large with respect to -— -, this equation
d(jj
gives us approximately the following X£2Z.
and more exactly
I
, _ p I rfTj
,0 „ -- _ _ -^_. ,
In addition, we have
Ia>'' _ /)' -P rfu7 I /rfT, y
2 2 I 1 2 1 \ (?lo' /
rfT,
We thus obtain
In the second member, the term before the last is a constant. The
last term is negligible, because I is very large.
Since we may add an arbitrary constant to H' without changing the
Hamiltonian principle, we may set
H'=T,-^U
and we know that the integral
J"= Cwdt
must be minimum (even though the initial and final values of u) are not
given) .
In the expression of H" , oi' must be regarded as a given constant.
H" is then a quadratic function, which is not homogeneous with respect to
the Xj^'s, having the form Hq + Hi + H2.
For example, let a material point having the mass 1 move in a
plane, whose coordinates with respect to the moving axes are 5 and n.
We shall have
We therefore have
260
The integral
is then minimum, when we assume that the limits tg and t^ are given, /260
as well as the initial and the final values of 5 and n.
The energy integral may then be written
and we have seen that the integral
is minimum even though we do not assume that tg and t^ are given.
We then obtain
i'==Ji2lh -.- II, ) dt ^f[ds /Ho -*- h + o.
The first term of the quantity under the sign J which is ds v/Hj + h
is essentially positive. This will "not be true for the second term,
which changes sign when we reverse the direction in which the trajectory
is assumed to be traversed.
If the point 5, n is very close to the border of the region in
which it is confined, and if, consequently, Hq + h is very small, the
first term will be very small, and the second term is the one which
will give the term its sign.
J' is therefore not essentially positive. This can also be seen
by means of the following equation
261
If h Is negative, the first term J is positive and the second is
negative.
Kinetic Focus /261
341. Up to the present, when I have stated that a certain integral
is minimum . I was employing abridged terminology which was incorrect and
could not deceive anyone. I should say the first variation of this inte-
gral is zero ; this condition is necessary in order that there be a minimum,
but it is not sufficient.
We shall now try to determine the condition for which the integrals
J and J' , which we studied in the preceding sections and whose first
variations are zero, are effectively minimum. This investigation is
related to the difficult question of second variations and the excellent
theory of kinetic focus.
Let us recall the principles of these theories.
Let xi, X2, ..., x^ be the functions of t; let x\, i^2» • • • . x^ be
their derivatives. Let us consider the integral
J =y '/(xi, x[)dt,
whose first variation 6 J is zero, assuming that the initial and final
values of the x^'s are given.
In order that this integral may be minimum, a condition which I
shall call condition (A) is necessary, but not sufficient.
The condition is that
/(Xi, X'i -I- E,) — Se/ -^ ,
regarded as a function of the Cj^'s, is minimum .
Condition (A) is not sufficient, unless the integration limits are
not very close. Except for this case, it is necessary to add another
condition which I shall call condition (B) . In order to explain this,
I must first recall the definition of kinetic focus.
In order that
262
^:J ^ o,
it is necessary and sufficient that the x^'s satisfy n differential /262
equations of the second order, which I shall call equations (C) .
Let
be a solution of these equations.
Let us set the following for an infinitely close solution
and let us formulate the variational equations, the linear equations of
which satisfy the d's and which I shall call (D) .
form
The general solution of these equations (D) will have the following
i = J/i
ii=J^^khk (i = i, a, ..., rt).
The Aj^'s are 2n integration constants, and the S^k'^ are 2n^ func-
tions of t, which are determined perfectly and which correspond to 2n
particular solutions of the linear equations (D) .
Under this assumption, let us state that the ^^^'s all vanish for
two given times t = t' , and t = t". We shall have 2n linear equations
between which we may eliminate the 2n unknowns Aj^..
We shall thus obtain the equation
where A is the determinant
Sn.i td.i
SI. lit
?n.t ?/»> ••• in.tn I
The quantities C'^j^ and S^j^ represent that which the function E,^y^ becomes
when t is replaced by t' and by t".
263
If the times t' and t" satisfy the equation A = 0, we may say
that these are two conjugate times and that the two points M* and M"
in space having n dimensions, which have J_lb3^
■•?i(''). ojC.'') ?'.(f).
respectively as coordinates, are two conjugate points .
In addit
is the close
ttion, if t" is the time conjugate to t' after t', which
2St to' t' , we may state that M" is the focus of M' .
We may now state the following condition (B) : There is no conju-
gate time of to between to and t\.
In order that J be a minimum, it is necessary and sufficient that
the conditions (A) and (B) be fulfilled.
A direct consequence may be inferred from this.
Let to, ti, t2, t3 be four times.
Let Mq, Ml, M2, M3 be the corresponding points of the curve
Let us assume that M^ is the focus of Mq and M3 that of M2.
If condition (A) is fulfilled, we may have
or
or
But we cannot have
Otherwise, the integral
hh
readily seen.
A curve in space having n dimensions
Xi -- 0,(0
representing a solution of the equations (c) can be called a trajectory,
which I shall call (T) .
The curve
Xj — o,- ^- li
will represent an infinitely close trajectory.
If we draw one of these trajectories (T' ) which are infinitely close
to (T) through the point M' , and if this trajectory again intersects the
trajectory (T) at M" (more precisely, the distance from M" to this tra-
jectory will be an infinitely small quantity of higher order), the points
M' and M" will be conjugate if, in addition, the point which follows
(T') passes through M' and infinitely close to M" at the times t' and t".
342. In the case of the Kamiltonian principle, condition (A) is
always fulfilled. In effect, we have
II = H(,+ H,-^H„
and H2 is a quadratic form which is homogeneous with respect to the xL 's.
In all of these problems of dynamics, this quadratic form is definite
and positive.
If we change xj^ into x'^ + e^. Hi will change into
H,(.;)-.s.g
and H2 will change into
265
dUt
and in addition we have
Therefore , we have
from which we finally have
7265
The first term corresponds to the function
Since the quadratic form E^ie^ ^^ positive definite and we may see
that the expression is minimum for Ei = - i.e., that condition (A)
is fulfilled.
343 Let us proceed to the case of the Maupertuis principle in
absolute motion. The integral to be examined may then be written
where dx^ is a positive definite, quadratic form with respect to the
differentials dx-j^.
For the time being, let us select xi as the independent variable,
The integral becomes
J dx,
dT\
„here l^^ is a polynomial of the second order P which is not homo-
Y dx]^ / dx-j^
geneous (but essentially positive) with respect to the -j^'s. Therefore,
let us set
d-z
dx
\-\/^^'
266
We must determine whether
\/"(i;*'.)--';^-'i^
Is minimum for e^ = 0. In other words, we must determine whether the
second derivative, with respect to t, of the radical
IS positive.
dxj ,
No matter what the -5 — 's and the e^'s may be, we shall have /266
P[~- -1- £,«) = af^ + 2bt + c,
\
- -1- e,n = at
■1 /
where a, b, c are independent of t. The second derivative of the radical
then equals
ac — 6»
(«<' -h 2bt +- c)'
Since the polynomial P is essentially positive, this expression is also
always positive, and condition (A) is always fulfilled.
344. Let us proceed to the Maupertuis principle in relative motion.
We must then consider the integral
J [ds /U„ 4- h + a;'(J dr, — tj d'^)],
or, choosing E, as the independent variable, we have
J ; [ v/(llo-4-/i)(7TV'J + <"'($'■/ — r, )] .
We must therefore determine whether the second derivative with respect
to n' of
V^( Ho + /0(H- T,'«) - 0>'( Jt/- 7))
is positive. This derivative is
Condition (A) is therefore always fulfilled.
Thus, condition (A) is itself fulfilled in every case which we shall
267
examine ,
Maupertuis Focus
345, The kinetic focuses are not always the same, depending on
whether Hamiltonian action or Maupertuis action is being considered.
In order to clarify this point, let us assume only two degrees of free-
dom, and let x and y be the two variables which define the position of /267
the' system, and which we may regard as the coordinates of a point in
a plane.
Let
^-/.(O, JK=/,(0
be the equations of a trajectory (T) which will be a plane curve. Let
us set
and, neglecting the squares of C and of n, let us formulate the varia-
tional equations. Since they are linear and of the fourth order, we
shall have
t; = a, j;i + rtjr,, 4- ajT,, + n^r^i,
where the aj^'s are integration constants, and the ^i's and n^'s are
functions of t.
The equation given in No. 341
may then be written
w r, r, w
T.'i T.'j -''% li
£" t' f* f
-.1 ?j ?3 ■:»
I'm T,I ^13 T,j
(1)
It is this equation which defines the Hamiltonian focus .
It indicates that the point x, y, which describes the trajectory
(T), and the point x + C, y + n, which describes the infinitely close
trajectory (T'), occur at two different times, i.e., at the times t' and
t", separated by an infinitely small distance of higher order.
268
However, these are not the conditions which the Maupertuis focuses
must fulfill. Two points of the trajectory (T) — i.e., the two points
M and M" which correspond to the times t' and t" — must be separated
by^an infinitely small distance of higher order from the trajectory
(T'). However, it is not necessary that the moving point which tra-
verses (T') passes precisely at the time t" — for example, infinitely
close to M". On the other hand, the energy constant must have the same
value for (T) and for (T' ) . This last condition is not imposed on
Hamiltonian focuses.
One of the solutions of the variational equations is /268
We may therefore assvmie that
The two functions C^ and m are thus defined.
In addition, the difference between the energy constant relative to
(T) and the energy constant relative to (T' ) is infinitely small. This
is obviously a linear function of the four infinitely small constants
Sl» ^2, 33, at^.
Without limiting the conditions of generality, we may assume that
this difference is precisely equal to a^.
The condition stipulating that the value of the energy constant be
the same for T and (T' ) is then ai+ = 0, or
For t = t', C and n must be zero, from which we have equations
«j y,\ + a, T,', + a, T,', = o.
In addition, the value of x + 5, y + n for t = t" + e must be the
same (up to quantities which are infinitely small of a higher degree)
as that of x and y for t = t", which may be written
(£ + a,)?; + a,fJ + a, tJ = o,
(£ + a, )t,; -t- (7,T,; -4- a,Ti; = o,
from which we have, by elimination.
269
i; k, r. o
'ii l', t/, o
By developing the determinant, we obtain
sj'ii ?:'ii ii'u — Cj^ii
(2)
and, setting
7269
equation (2) becomes
= a'),
?(«') = !:('';•
(3)
Application to Periodic Solutions
346 If we are dealing with a periodic solution of period Ztt, the
functions fi(t) and f2(t) of the preceding section will be periodic of
the period 2tt. The same holds true for
In addition, according to Chapter IV, the variational equations will
have other particular solutions which will have the following form
In these equations, g is a constant, a and -a are the characteristic
exponents, and the t', it is necessary that
k be positive.
.11 4-1
By setting k = 1, we shall give the smallest value to t" - t'. We
have
T ^ T = — -
and the point M" is then the focus of M' .
One factor must be pointed out.
In order that the preceding line of reasoning may be applicable, it
is necessary that log G(t) be a periodic function. However, in general,
all that we know is that G(t) is a periodic function, and as a result
logGC/)
is increased by a multiple of 2iTr, for example, of 2ki7r, when t increases
by 2iT. Then
logG(0 — I*'
is a periodic function.
Let us then set q.^,) ^ q^,)^-,*-/^
a = a H ;
2
we have ^(,) = g,„G(0 = e"''G'(0-
We shall then no longer set
I
21
272
but rather
Since log G(t) will be periodic, the preceding conclusions remain valid,
and equation (3) will be written
T — X = — — (m is an integer number)
and, in addition, M" will be the focus of M' if /212
7.
348. One of our three hypotheses stating that log G(t) must be
periodic has thus been proven, I may now state that the function t
must be constantly increasing, as we assumed.
Let us assume that this function has a maximum tq for t = tg. We
may then find two times t'^ and t'j* such that the corresponding values
x\ and Tj of the function t are equal, and two other times t2 and t^^
such that Tj = T^ and such that the five times which are very close
to one another satisfy the following inequalities
Then t" will be the focus of t' , t" that of t\. We saw above that
such inequalities are impossible when condition A is fulfilled.
I may now state that G(t) cannot vanish. We have
The numerator and the denominator of c(t) are imaginary and conju-
gate. If one of them vanishes, the other also vanishes, so that the
function ?(t) cannot become either zero or infinite.
Thus, all of our hypotheses have been proven.
Unstable Solutions
349. Let us now assume that the unstable solution and a^ are
positive; in this case Sa. 12. ?3, na, ?, a, G are real.
For the same reason as given above, the function t will be con-
stantly increasing. However, two hypotheses are possible:
273
1. The quantity c(t) cannot vanish nor become Infinite , and
Increases constantly from to +«> when t increases from -» to +°°,
It then happens that no point of our periodic solution has a
Maupertuis focus .
2. The quantity C(t) may vanish for t = to- It will also vanish /273
for t = to + 2Tr, and since It cannot have either a maximum or a minimum
it must become infinite In the Interval. In the same way, if C(t) can
become infinite, it must also be able to vanish.
In order to clarify our thoughts, let us assume that c(t) becomes
infinite for
and for values which differ from these by a multiple of 2Tr, and vanishes
for
I shall assume that
h.
The closed trajectory (T) which represents our periodic solution
will therefore be divided into two arcs, whose extremities will corres-
pond to the following values of t
Each of the points of one of the arcs will have Its first focus on
the following arc.
I may add that the points corresponding to the values of t
'oi 'o' 'l> 'l
coincide with their two focuses.
Let t" be a value of t corresponding to an arbitrary point of (T) ,
and let t^ be the value of t which corresponds to its niS. focus. We
shall have
lim =
n 1
Ilk
However, this is not all; we shall have
If n is very large and if G(t") is not infinite, since t^ - t" is very
large and since we assume that a is positive, G(t") will be very /274
small, so that if t" is, for example, included between tg and ti, the
difference
will strive toward t' when n increases indefinitely.
If n strives toward -°°, this difference will strive toward tg or
toward t^ , depending on whether t" will be Included between tg and t'o
or between tVand t^. I should add that the difference f'jj, - 2nTr is
either constantly increasing or constantly decreasing with n.
The values t q , t' correspond to the points where
but 51^2 ~ ?2^1 ^s ^ periodic function multiplied by e*^ . However, a
periodic function must vanish an even number of times in one period.
Consequently, the closed trajectory (T) will be divided by the
points to, ti, tg + 2-n into a certain nxjmber of arcs, and this number
will always be even .
350. From the point of view in which we are interested, the un-
stable, periodic solutions may be divided into two categroies. However,
it could be asked whether these two categories exist in actuality. It
is therefore advantageous to cite some examples.
Let p and w be the polar coordinates of a moving point in a plane.
The equations of motion may be written
For p = 1, let us assume that we have
t/w dp ' dp' '' '^ •
Equations (1) will have the solutions
and this solution will correspond to a closed trajectory which will be
a circtnnference.
275
K^
Let us set
and let us formulate the variational equations. They may be written
The second may be integrated immediately
dt ^
but this constant must be zero if we want the energy constant to have
the same value for the trajectory (T) and for the infinitely close tra-
jectory.
Therefore, if we replace -^ by -2c, the first variational equation
will become
;5J| = a-f(0-3]. (2)
Equation (2) which remains to be integrated is a linear equation
having a periodic coefficient.
These equations were discussed in Sections 29 and 189 (see, in
addition. Chapter IV, in various places) .
It is known that they have two solutions of the following form:
where G and Gi are periodic functions.
We are going to present examples for every case mentioned above.
Let us first assume that ^ may be reduced to a constant A (case of
central forces) .
If A < 3, we shall have a stable, periodic solution.
If A > 3, there will not be a Maupertuis focus on (T) , and we
shall have an unstable, periodic solution of the first category.
I must now show that we may also have periodic, unstable solutions
of the second category.
The solution will be unstable and of the second category if G
vanishes in such a way that the ratio
276
1115
which corresponds to the ftinctlon ^(t) of the preceding sections can Jllh
vanish, and consequently can become infinite.
We may obviously formulate a periodic function G which satisfies
the following conditions:
1. It has two simple zeros and only two;
2. These zeros will also vanish
if G dC,
dl'- dt
As a result, every time that
vanishes, its second derivative will also vanish in such a way that the
ratio
remains finite.
One could obviously formulate a function G which satisfies these
conditions. The periodic function ({> formulated by means of this func-
tion G will correspond to an unstable, periodic solution of the second
category.
As an example of function G satisfying this condition, we may set
G — sliif— - (cost — cosSO-
4
This fxmction vanishes for t = and t = ir, and it does not have
another zero if
/a
For t = and for t = tt, we have
di'-^'-^dT--''-
Q
In order that the ratio -r— may vanish, it Is not sufficient that
G vanish;, it is still necessary that G^ does not vanish.
277
\^
However, this is what occurs, because if G and Gi vanished at
the same time, the two solutions
/277
'G(0,
<,'-^'G,(0
could only differ by a constant factor (since they satisfy the same
differential equation of the second order), and this is absurd.
351. One point to which I would like to draw attention is the
fact that the unstable solutions of the first category and of the
second category form two separate groups, so that we cannot pass from
one to another continuously without passing through the intermediary
of the stable solutions.
Let us first confine ourselves to the particular case given in
the preceding section, and let us reconsider the equation
3).
(2)
Let us vary the function <() continuously, and let us determine whether
we can pass directly from an unstable solution of the first category to an
unstable solution of the second category. For this purpose, it is neces-
sary that the function G, which is real, be first incapable of vanishing,
and then be capable of vanishing. We would thus pass from the case in
which the equation G = has all its imaginary roots to the case in which
it has real roots. At the time of passage, it would have a double root
or, more generally, a multiple root on the order of 2m.
This zero, which would be on the order of 2m for G, would be on the
■ of
presslon
order of 2m - 1 for ■^, on the order of 2m - 2 for -^, so that the ex-
Tin"'
2 s -; \- r-G
dl
would be come infinite, which is impossible since it equals '-r/^'). (7)
... . d^U _^ d^U ,
I shall designate the sum of the two second derivatives — -y + — 2 ^y
AU (or more briefly by A) .
The following identity may be easily verified
= 1{/ x' ~ y x')(\ x' ^ r^ y -\x' -- -ry),
or, taking into account (5), (6), (7) and (3), we have
(yj_,-_y'i)(6'- 6i)- 2(x'a:' + y7')0' -H 2(x"» + y )0 ^ o. (8)
This is the differential equation which defines the unknown function 9.
We shall set
and our equation becomes
282
X — 2 ui r = — , y + 2 w a:' = -j— ,
ax ay
(9)
an equation having the same form as equation (2) of the preceding section.
The conclusions of the preceding section therefore remain in force. One
periodic unstable solution is of the second category, or of the first
category, depending on whether the function i) can vanish or not. We can-
not pass directly from an unstable solution of the first category to an
unstable solution of the second category, but can only pass through
stable solutions.
354. Do the same results still remain valid in the case of relative
motion?
The equations of motion then become
d\i . , dM (1")
dx' ^+---=^.
where o) designates the speed of rotation of moving axes.
The variational equations will be /283
Due to the fact that the energy equation is still valid, the same will
hold true for
Let us set
and equations (5) and (6) will continue to hold.
In addition, since x' and y' must satisfy equations (2'), we shall
have
, ■^'U , d^M ,
rf'U , rf'U ,
Taking these equations into account, as well as equations (2'), and also
taking into account equation (3) , we may simplify the expression of 6",
and we again obtain the equation
283
Since the identity given in the preceding section is always valid,
we shall obtain equations (8) and (9) again. Therefore, nothing needs
to be changed in the conclusions given in the preceding section.
355. However, one new question arises.
The trajectory (T) is a closed curve. Up to the present, we have
tried to determine whether an arc AB of this curve would correspond to
an action which is smaller than any infinitely adjacent arc with the
same end points.
However, we may also question whether this entire closed curve
corresponds to an action which is smaller than every infinitely small
closed curve.
Let us first assume that a point A of the curve (T) has its first
focus B on the curve (T) , so that the arc AB is smaller than the entire /284
closed curve.
This is what occurs for unstable solutions of the first category.
We have seen that the curve (T) may be divided into a certain even number
of arcs for these solutions, and that every point on one of these arcs
has its first focus on the following arc, so that — starting from an
arbitrary point — its first focus will be encountered before the entire
curve (T) has been traversed.
This also occurs for certain stable solutions. In the case of
stable solutions, we have set (No. 347)
and we have seen that the x of a point, and that of its first focus,
differ by — . Therefore, if y is larger than y, the focus of a point
will be encountered before (T) is completely traversed.
If this is the case, the action cannot be less for the curve (T)
than it is for any Infinitely adjacent curve.
Let ABCDEA be the curve (T) , and let us assume that D is the focus
of C. Since E is outside the focus of C, we may attach C to E by an arc
CME which is very close to CDE, and which corresponds to a smaller action.
If I represent the action corresponding to the arc CME by (CME),
we shall have
284
and, consequently,
(CME)<(CDE)
(ABC.MEA)<(ABCDEA).
Let us now consider a stable solution, such that
It may be stated that the action will no longer be less for (T) than
it is for any infinitely adjacent closed curve.
In order to clarify these ideas, I have compiled a figure, assuming
a 11
that — ranges between -r and 7-, in such a way that the focus of a point
is encountered before traversing (T) three times, and after traversing /285
(T) twice.
Let ABCDA be the curve (T) . The focus F will be located between
AB, and it will be encountered after traversing (T) twice.
Since B is located beyond this focus, we may attach A to B by an
arc AEHNKHMEB, such that
(Ai;iIMaiMl-:B)<(ABCABC\B).
Since the focus of A is not encountered by describing the arc AB without
Figure 10
traversing (T) , we shall have in addition
(AE + EB)>(AB),
285
v-^
from which we have the following by subtraction
(F,IL\KHME)<(ABC.\BCA)
°^ (EII.ME)-+-(H.N'KH)<^.{ABCA).
We must therefore have either
(EII.MK)<(ACGA)
°^ (IIMvIIXCABCA),
In every case, there is a closed curve which differs little from
(T) and corresponds to a smaller action.
Therefore, in order that a closed curve may correspon d to an action
which is less than any infinitlv adiacent closed cu rve, it is necessary /286
that this closed curve correspond to an unstable, peri odic solution of
the first category .
356. Is this condition sufficient? In order to determine this,
let us study the asymptotic solutions corresponding to a similar un-
stable, periodic solution.
Let
a- = ?o(' , y = 'WiO
be the equations of the periodic solution, and let
X = 00(0 + Ae^'9i{/)H- A'e"''fj(0-t-- • •,
^.-=^„(04-Ae^'|,(0-+-A«c'-'.^,(0 + ---
be the equations of the asymptotic solutions. The functions {A,B,).
We therefore have
(CB3)>(A,B0,
and, consequently,
action of (T') > action of (T) .
q.e.d.
358. We must now determine whether the same result is still obtained
for relative -motion.
The irreversibility of the equations constitutes a great difference
289
from the preceding case. The action for an arbitrary arc AB is no
longer the same as for the same arc traversed in a different direction.
If an arbitrary curve satisfies differential equations, this will not
hold true for the same curve traversed in a different direction.
Finally, the orthogonal trajectories of the curves CD will no
longer have the basic property which I disciissed in the preceding
section. However, there are other curves which I shall define, and
which have this property. This is sufficient for the result of the
preceding section to remain valid.
In No. 340, we obtained the following for the expression of the Z290
action
J'= f[ds v/H7+A + lo'(? c/t, - T^ d';)].
For purposes of simplification, I shall set V/Hq + h = F. I shall no
longer designate the coordinates by C and n, but rather by x and y, in
order to approximate the notation employed in the preceding sections.
And I shall no longer designate the angular velocity by u' , but rather
by 0), removing the accent which has become useless. I shall then have
J'=: f[F '\' dr d-r dy d'y
ds ds' ds ds' '
we arrive at an identity.
If we therefore consider the curves (1) , they will satisfy equation
(6). If we take this equation into account, relationship (4) becomes
^ ,, r,, dx 5.r i- dy iy . , 1 '
Let AjBi, A2B2, ..., A^Bn ^^ ^ continuoiK series of arcs pertaining to
the curves (1), whose end points kik2. . .k^, BiB2...Bj^ form two continuous
curves C and C .
Let AiBj^, Ai+]^Bi+i be two of these arcs which differ from each other
by an infinitely small amount. Let x, y be the coordinates of the point
A^, X + 6x, and let y + 6y be the coordinates of the infinitely adjacent
point A^+2 .
Let J' be the action relative to the arc A^^B-j^ and J' + 6J' be the
action relative to the arc Aj^+iBj^+j.
If a is the angle which the tangent to the curve A-j^B-j^ [which is a
curve (1)] makes with the axis of the x's, and if the two curves C and C'
satisfy the differential equation
r'(coso! 'jx -\~ sin a 5_k) -t- (0(0: S/ — y ox) =0, (7)
we shall have
8J' = o
291
and , cons equent ly ,
(A,B,)..(A5nj)=.... = (A„15„)-
The curves defined by equation (7) may therefore play the role which
the orthogonal trajectories of the curves (1) played in the preceding
section.
We may therefore consider figure (2) again, and we may assume /292
that the curves shown by the dot-dash line no longer represent these
orthogonal trajectories, but rather the curves defined by equation (7).
There will be nothing to change in the proof.
However, one point is no longer clear. In the infinitely small,
rectangular triangle A3CB3, I have
(CB,)>(.\,B,).
The triangle is no longer rectangular, and in addition I have changed
the definition of the action. Does the inequality still exist?
It may be readily seen that this equality equals conditions (a) of
No. 341, and we have seen in No. 344 that they are fulfilled. The in-
equality therefore holds, and our proof remains valid.
To sum up, in order that a closed curve corresponds to an action
which is less than any infinitely adjacent closed curve, it is necessary
and sufficient that this closed curve corresponds to an unstable, periodic
solution of the first category .
359. We must make a few remarks regarding the classification of
unstable solutions into two categories.
From another point of view, the unstable, periodic solutions may
be divided into two classes. Those of the first class are those for
which the characteristic exponents a is real, so that e^^^ ^g real and
positive, where T is the period.
The solutions of the second class are those for which this exponent
a has "Y as an imaginary part, so that e^T is real and negative.
In the preceding statements, we only considered unstable solutions
of the first class. Let us see whether those of the second class may
also be divided into two categories. .
We may set , , i'^
292
where a' is real. We may then set, just as in No. 346"
where (^2, *2. (,-i-e, -Hs'-tj-t-
I would like to point out that ^y^ is homogeneous of degree k + 2,
if we assume that Cp and u^ are of degree p + 2, t\ and v are of the
297
KJ
degree p, X and v^ is of degree p.
It is therefore a whole polynomial with respect to
^p. Up, T^P, <>. '^P< PP (/>>0)i
and with respect to
/;(„«"<, /"
ofi"
These last two quantities may be assumed to be on the order of 1.
Finally, the coefficients of this polynomial are periodic functions of
no whose period is 2v .
In addition, we shall obtain
where Hq and Bq are the values of ^ and ^ for X = y = 0. (We may
assume that we have -^ = -^ = for A = y = 0.) In addition, 0^ depends
only on
Ip, -Tip, Up, Vp, Ip, Up ip^k — i).
Our differential equations may then be written
/299
'iT^'J^,' dt'^ di,' dt ch, dt du.
For k = 0, they may be reduced to
d\i, _ rff'j _ . lAo _ . dva _ .
llT ~ ~dt ~ ° ' dt "'' ~dt "^ "*'
They demonstrate the fact that Co and ^^O are constants, and that
7)0 — t-i vi= int + EI,
where oT is a constant which must be determined.
We may advantageously add other equations having a similar form to
equations (4) and (5), which are only transformations of them.
Let us develop xi and yi in powers of e, and let
^i-ro + ^ri +^'li +■■•- (4')
The expansions (4') may be directly concluded from the two last ex-
pansions (4) .
We then find that ^-^^ is a whole polynomial with respect to the
298
quantities
?p. ^p, \'p, 1^. '^p, \^p (writing Hq separately) , (6)
and that this polynomial is homogeneous of degree k + 2, if we assume
that
?p is of degree p + ■».,
?p, r^'p is of degree p -hi,
>■,/). ^p> \^p is of degree p.
We then have the following equations
'l\\. ^ "+?'''^?='?,
dt ' dt '
and we shall derive the following
^ ^ii^p-h qn)
^ _ ^ . r gf(p*^-?^<+JCTl_|- Y'e-'"',
' j^i(.p-\- qn-^ n)
where y and y' are integration constants.
Therefore, if X and Y are whole and homogeneous polynomials with
respect to
the same will hold true for 5 and n, unless it is assumed that the con-
stants Y and y' are zero. If it is not assumed that these constants are
zero, K and n will still be whole polynomials, but not homogeneous.
Let us apply these principles to the quantities which we have just
computed. Due to the fact that
302
dSt rfSi rfd| dOi
rf;o ^1o rf;i rflj
are polynomials which, according to the convention which we have em-
ployed regarding degrees, are of the following degrees, respectively
1, 3, 1, 2,
the same will hold true for
III ?l. ')l> ?!•
When we substitute the values of these quantities which are, re-
spectively, of degrees 1, 3, 2, 2, instead of these quantities in'e2,
it may be seen that 02 becomes a polynomial of the fourth degree, and /304
that -^
rfe, de, de, de,
d^o' dTTio' d^',' rfT)J
will be polynomials of the following degrees, respectively
2, 4, 3, 3.
We may therefore formulate a generalization of this result.
Equations (5) and (5') enable us to compute the unknowns 5, , t\,,
C\c> n|^ from place to place. This would only be prevented if the mean
value of the second term of one of the equations (5) were different from
zero.
Let us assume that this does not occur. It may be stated that
b-< T)i, ^i, r/i
will be polynomials of the following degrees
X + 2, X-, k+l, k + l
with respect to
where the coefficients of these polynomials are themselves periodic func-
tions of t of period 2Tr.
Let us assume that this is valid for every value o,f the index which
is less than k.
We know that 0j^ is a whole polynomial of degree k + 2 with respect to
??• ';?> i'j, ^I'q (?<*) /■^2_)
303
\^
assuming that these quantities are of degree q + 2, q, q + 1, q + 1»
respectively. If we substitute polynomials whose degree, wxth respect
to the quantities (10), is precisely q + 2, q, q + 1, q + 1, in place
of these quantities (11) , it is apparent that the result of the substi-
tution will be a polynomial of degree k + 2 with respect the the quan-
tities (10) .
Therefore, 0], is a polynomial of degree k + 2 with respect to the
quantities (10) , and for the same reason
den dOk dQk dSj
will be polynomials of the following degrees
k, k-hi, k-\-l, k-hl
with respect to the same quantities.
The same holds true for the second terms of the first, second,
fifth, and sixth equations (7). Consequently, by repeating the previous
line of reasoning, we should readily see that the same holds true for
/305
T,*, U■^ ^'kl I'll-
q.e.d.
The integration of equations (7) has introduced four new integration
constants. They provide us with information concerning 5i, ni, ?i , ni,
up to the following terms
containing the four arbitrary constants
Yi, Si. Yi' S'l-
We shall retain only one of these constants and we shall set
Y, = 8i = o, S', = — y',.
Under this assumption, let us try to determine
?I, 1,, J',, T|',,
by means of equations (5) and (5') and by setting k = 2.
It is necessary that the second term of the first equation (5) has
a mean value of zero. This mean value equals
[S]'
304
and we always employ the brackets to represent the mean value of a
function. We must therefore have
m-
(9')
Let us assume that 02 is developed in Fourier series in the following
form
Since 62 is a polynomial of the fourth degree, q could not exceed 4 in
absolute value. Consequently, if the denominator of n is larger than /306
4, [©2] will be identically zero, and the condition (9') will be ful-
filled by itself. The constant u will remain undetermined.
If the denominator of n equals 2 or 4, the condition (9') will de-
termine TU.
If the denominator of n equals 3, the constant "S has already been
determined by condition (9), and condition (9') will enable us to deter-
mine the constant y'l •
Let us calculate the terms depending on this constant y\ in 02.
We obviously will obtain
Yi rfe,
i.e.
The mean value of this will be
The condition (9') may therefore be written { if it is noted that
_lL ^'fQ.I ^ H - „
where H depends on oT, but not on ■/, .
If the denominator of n does not equal 3, [Gj] is zero and condition
(9') is independent of y\. Therefore, if this denominator equals 2 or 4,
305
equation (9') will depend on w and not on fi and will determine o).
If the denominator equals 3, condition C9') depends on y'l and will
determine y' (it will provide us with y\ = 0) .
In any case, having thus determined £.2, let us try to calculate n2
by means of the second equation (.5). We shall employ X2 in such a way
that the second term has a mean value of zero.
We should point out that X2 will not be zero in general, and
will not be zero in general, because, due to the fact that 02 is a poly-
nomial of degree 4, it will include a term containing Cg which is inde-
pendent of the Cy.'s and n'^'s. The coefficient of this term will be a
periodic function of t of period 2tt, and the mean value will not be zero
in general.
Let us proceed to equations (5') or, which is the same thing, to the
last two equations (5). The second terms of these last two equations must
have a mean value of zero.
/3Q7
We must have
[£]=--■'••
which determines v^' However,
de, , f/B, M,
Mj -3 — — ^lo "7JT -+- ?o ~J~r
dua d^fi dr^^
is a polynomial of the fourth order. F2 therefore includes terms containing
x?v?. and consequently U2 —^ includes a term containing
^^ dUQ
III = (/^e''"*=')HV'«^e-'""^ra))'.
The coefficient of this term is a periodic function of t, whose mean value
is not zero in general. Therefore, in general L_i and, consequently.
y2 are not zero. This is the same line of reasoning as is employed for
'2
A2.
We must then have f de,!
[d^i~°^ (12)
306
However, it may be stated that this condition is fulfilled by itself.
We have the energy integral F = const., from which we may deduce the
series of equations
'J'o = const., , = consl., *j = const.,
Let us consider the third of these equations
*i = 6j— ;»— inui + ^iIlo?o + 2B(i[jii«(i = const.
This equation may replace the fourth equations (5) and, when X2, ^2. ^2 . /308
TI2 and v„ have been determined by means of the first three equations (5),
it will determine U2 without any integration . We may therefore be assured
that the determination of U2 is possible, and, consequently, that the con-
dition (12) is fulfilled.
We will have thus determined C2. ^2. C2 . n'2 up to the following terms
depending on the four arbitrary constants. We shall retain only one of
these constants, and we shall set
363. The calculation will be continued in the same way. The ability
of equations (5) to be integrated requires the following conditions
The last two conditions will determine X^ and yj^. The second will be a
consequence of the first, according to what we have learned with respect
to condition (12). We must then study the first.
dGk
Expression -^ is a polynomial of order k + 2. If it is developed
in Fourier series
the integer number q cannot exceed k + 2 in absolute value. If k + 2
is smaller than the denominator of n, we could not have
p -i- gn =zo
and the mean value of our expression will be zero. The condition
L5^J=** (13)
307
will therefore be fulfilled by itself.
We have introduced the following arbitrary constants:
n, Y„ Y„ ... (1^)
^ J /309
and Oy^ may depend on -^
'^' Yi. Yi> ■■■• Y*-i-
Let us determine the form of this dependence. Let us assume that we are
considering the expansion
and that in this expansion we replace the 5's, the n's, the C -s and the
n' 's by their values. The different terms of the expansion will then de-
pend on the constants (14) . In this expansion (15) , let us cancel all the
constants y' , retaining only aJ, We will thus obtain a new expansion
*;+£*', + £'*', -1- (16)
In the expansion (16), let us now replace the constant IB by the expan-
sion
ra -t- EtTTi + e» Oj -V- . . . ,
where wi , W2 are new constants. Each term in the expansion (16) may be
developed in its turn in powers of e. When this expansion is ordered
anew in powers of e , we obtain a new expansion
This expansion must be identical to the expansion (15) , under the condition
that the constants u are replaced by the suitably chosen functions of the
constants yl.
It may be readily seen that $j^ depends only on
O, TO,, ..-, ^k-i
and that \ depends only on
ra. Yi' •■■' If'*-''
We may conclude from this that To depends only on
Yi. Yi' •••' f*
and y'^ on ^,_ ^^^ ..., n,*.
308
It may be readily seen that /310
t>\ = SAD*;,, Ts^ raj- . . . Bjjt,
where A is a numerical coefficient and where D$' is a derivative of $'
_ mm
with respect to w. The order of this derivative equals
"i + if-h. ..-hat
and we then have
k = m + tit-h2Xt-h...+ /c-xt.
Since m is at least equal to 1, and since "fo does not depend on ¥,' it may
be seen that aj^ is zero, which we already knew.
Let us consider an arbitrary term where Oj^,, a^^^i, ..., ct, ,, are zero,
but where a^ is not zero. We must have
rngfc-h.
If the denominator of n is larger than k - h + 2, the mean value of D$
will be zero. This means that those terms of 4>|^ which depend on IS^ have
a mean value of zero.
An important result may be concluded from this concerning the mean
value of $1^, and consequently the mean value of 0. .
If the denominator of n equals k + 2, [©k ] will depend only on w.
If the denominator of n equals k+1, [0j^] will depend on IS and oTi .
If the denominator of n equals k, [0j^] will depend on To, ui and 702.
_If the denominator of n equals k - 1, [6^^] will depend on IS, TSi , -©2
and 003.
The statements which I have Just made concerning [Qi,] also apply to
j-dG^I ^
UnoJ ■
Therefore, if the denominator of n equals k + 2, relationship (13),
which will only include w, will determine oT.
If the denominator equals k+1, relationship (13) will contain W
and 732 . However, "io will have been previously determined by the relation-
ship
309
^-/
Relationship (13) will therefore deterniine ui and, consequently, Vi-
If the denominator equals k, relationship (13) will contain w, a^i
and 11^2. However, Hi and UJ, will have been previously determined by rela- /311
tionships having the same form as (13). Therefore, (13) wxll determine
1:^2 and consequently Vs- This process will then be continued.
Discussion
364. The solution which we have obtained still includes the following
arbitrary constants
With respect to the parameters A and p, we have obtained them from
their expansions in increasing powers of e, and we have successively cal-
culated the coefficients of these expansions. These coefficients X^ and y,,
depend on the two constants Co and uq ; these coefficients were calculated
by means of the following equations
where O, , ^ and uq 4^ are whole polynomials in
k' dCo 6i-+- A, — 6, — A, > denominator of /2>2,
which indicates that R is divisible by uq , since uq is included with the
exponent - (bi + hi + ba + ha).
There would only be an exception to this if we had
b,-h A,= b,-hh^
but we would then have either
in such a way that R would be divisible by uq , or
b,— hi = bi = ht = o,
from which we have
A, -+■ A,
= o.
a
However, the corresponding terms would not then appear in un — ^ .
In the same way R will always be divisible by E,q , unless hs = 0, in
which case the term would not be included in E,q \—^\ .
/312
311
Therefore, to sum up we have
\den [fife*-]
\du,\' L'/uJ
and, consequently, \y, and m^ are whole polynomials of Cq and \^. There-
fore X and y are series which may be developed in powers of Z313
but these three constants do not enter arbitrarily.
Let us recall the method which we employed to introduce the auxil-
iary constant e, which only served to simplify the discussion. For this
purpose, let us again consider the notation given in No. 274, and on page
95 . We have set
X, = EX'j, >'i=Ey,, X, = E'xi, >'j=7i.
Therefore, our equations do not cease to be satisfied when we change
into
e*-i, x\k, y\k, x',A»
and when the parameters X and u retain their initial values.
We then remove the accents which have become useless, and we develop
x' , y', x', y', which we shall hereafter designate by the letters
x^', y ^^ y^2* ^1^ ^" powers of e. We thus obtained the expansions
( 'ii + er/j + Eir, ',+....
e
We shall not cease to satisfy the equations if we change e into j^,
and if we multiply the four expansions (19) , respectively by
*', I, *, k,
or, which is the same thing, if we change
5pi ^P' ^p' ^'P
^^^° l,k^-p, npk-p, lp*•-^ -,;.*'-''•
312
By means of this change, we must again obtain expansions which are
identical to the expansions (19), but with different values of the con- /314
stants Co ^^d un. However, it may be seen that Co and uq are changed
into k Co and k^UQ by means of this change.
Therefore
!Pi ''>pi Ipt 1^
change into
J/-*'-", ^pk-P, I'pk'-P, y^'^k'-P
when Co and uq change into k^Co and k^ug .
In other words , if the four expansions (19) are multiplied respec-
tively by E^, 1, e, e, the four products thus obtained may be developed
in powers of
The same must be true of X and y, which did not have to change when e,
Co, uq were changed into ^, k^Co , ^^^o-
Therefore, let us assume that A and y are expressed as functions of
e Co and gy/uq^ It is apparent that we shall thus have relationships from
which we may derive e^Co and ey^ inversely as functions of X and y.
365, Let k + 2 be the denominator of n. The constant US will then
be determined by the equation
There is only an exception to this in the case of k + 2 = 2, where "cj is
determined by
dOk
10
The expression -^ — is a whole polynomial of degree k + 2 with respect
to
Therefore, each of these terms contains factors having the form
Only terms which are independent of t will remain in the mean value
LdnoJ
313
and we have seen that q must be divisible by the denominator of n, i.e., /315
by k + 2.
Therefore, our expression has the following form
I shall now show that the coefficient b is zero.
For this purpose, I shall employ the following method. Let us cal-
culate
lo, ?ll •••' ?*■- 1'
■^0. ^i ^*-i'
So I Tl; •••• ?*-l'
To I III •••' '^*-l'
by the procedure presented above. However, when computing Cj^, I shall
retain an arbitrary value for C, instead of assigning a value which can-
cels [^^1 to u5. Then the following equation
LdnoJ
dt d7\a
Will allow me to compute ^k- However, instead of being a periodic func-
tion of t, ?k will be a periodic function of t in addition to a non-periodic
term
We have another method of calculating
5o, 5ii ■ • ■'' b)
lOl 1l> .•■■■ li— 1'
• • • 1 • • ■) '
and, consequently, this term t[.^J. This method consists of again per-
forming the calculation presented in No. 274.
We shall determine Sq, Si, ..., by means of equations (2) on page 100.
No difficulty will be entailed in calculating Sq, Si, ..., \.i, t)ut
we shall encounter some difficulty when calculating S^ by the equation
-r^ -4-2B -i— = *-t-G/.
dyi dv
In effect, the second term represents a group of terms having the ^^^^
following form
314
Ae'"'.r.+'"t'',
where m^ and m2 are integer numbers. Nothing impedes us from performing
integration, provided that we do not have
I'm, + 2mjB =0.
Since 2B equals in , where n is a commensurable number whose denominator
equals k + 2, the second term of our equation will include terms satisfying
this condition. As a result, Sj^ will not be a periodic function of y, and
V, but may equal
where Tj^ and Uj^ are periodic.
Having thus determined the fpctlon S and having obtained the approxi-
mation to terms of the order £^+ , we may employ the procedure given in
No. 275 and may thus determine Xj , y^ , X2 , y^ .
These two computational methods must lead to the same result. There-
fore, let us set
S = So-4-eSi-H... + E*S*.
Let us compile the equations (see page 102)
rfs dz ^ dz dz
dyi dv d(Xo d^o
dC dC
and let us derive X2 from them as a function of t. The value of x, which
is thus obtained must equal
?o-t-E?iH-...-He*^t
up to terms of the order e^"^l .
We are interested in calculating g^, particularly that of the sec-
ular term
m-
This secular term can only come from the secular term of S, , which eouals
We thus have the following, up to terms of the order £^+l (equating /317
the secular terms in the equation x„ = — )
2 dy2
315
\^
In the first approximation — i.e., up to terms of the order e — we
have (see page 102)
n, = i; nt=in\ mt -^ w^ = i> = Va= i{nt -^w).
We shall therefore commit an error of the order e^+'^ if, in the second
term of (20) , we replace
fo, Po, yi, p
by 5o> Mo, ', ((nt-hTa).
We shall therefore obtain h — J by making the same substxtution m
^^^ . However, % only includes terms containing
dy2
inixyt -+■ rritv,
Where im,^ ^m,B^-o.
We therefore have
—. = — 2 b — 7— •
dy, dv
dUk
However, \i\^ is a periodic function of yz and iv. Therefore, ^^
r^Qki
cannot contain a term which is independent of v. Therefore [^J does not
contain a term which is independent of w.
q . e . d .
In order to clarify the preceding calculation, I would like to make
one more remark. The mean motions nj and n^ are given by
dC dC
tl = — J— > Tli^ — -Jar'
aio d^o
In general, they depend on e, and they are only reduced to 1 and in
for E = 0,
However, we are here employing two parameters X and y which may be
replaced by the arbitrary functions of e, .or, if it is preferred, we may ^318
316
employ an infinite number of constants Ai , A2, ..., yi, y2» ••• • We may
then employ these constants in such a way that n^ and n2 remain equal to
1 and to in , no matter what e may be.
366. In order to determine "w, we therefore have an equation of the
following form
where a and c are conjugate and imaginary. In general, a and c are not
zero, otherwise H could only be determined to the following approximation.
^The equation will provide us with the following series of real values
for oj
■K III Stt
It is apparent that we do not have two values which are actually
different when we change oJ into u + Ztt, but we have more than this. It
may be stated that the two values
211
k ~t- 1
do not correspond to two periodic solutions which are actually different.
Since t is not explicitly included in our equations, by changing t
into t + h we may transform an arbitrary periodic solution into another
solution which is not essentially different.
Therefore, let us change t into t + 2hTT, where h is an integer number.
Then no changes into hq + 2hTr and vq = i(nt + w) into
Since all of our functions are periodic, of the period Ztt, in Hq and
iv, we shall not change our solution in any way by subtracting two multiples
of 2tt from tiq and — , respectively, for example 2hiT and 2h'TT . Then no will
again become no and vo will change into
i(nt -}- 2/iAit + xa — iA't).
In other words, we will have changed o) into /319
CI -+- iT.{nh — h').
However, we may always choose the integer numbers h and h' in such a
way that
317
^^
nh — A' = , , : •
A: -+- 2
We therefore do not obtain a solution which ia actually new by
- - 2Tr
changing co into u + j^ ^ 2 '
q.e.d.
We therefore have only two solutions which are actually different,
corresponding to the two following values of o)
TSa ■
/t^-2
We must now determine the constants z^E.^ and e^UQ. For this purpose,
we shall employ equations which relate these two constants to X and p. In
the questions which are customarily discussed, there is only one arbitrary
parameter, and we have introduced two in order to facilitate the discussion.
It is therefore convenient to assume that X and v are related by one rela-
tionship — for example, X = y.
The expansion of X and that of y in powers of £^^0 and ey^ begins in
general with terms containing e^^o and e^uQ (if we disregard the case m
which the denominator of n equals 3) .
If we therefore assume that y = X, we shall derive z^^q and e\/^
from this which may be developed in powers of \/T. Either the coefficients
of the expansion in powers of l/T will be real, or, on the contrary, the
coefficients of the expansion in powers of V^ will be the ones which are
real.
In the first case, the problem will have two real solutions for X >
and will not have any for X < 0. In the second case, the opposite will
hold true.
In order to determine which of these two cases is valid, let us examine
the equation which relates y to uq, restricting ourselves to terms con- /320
taining e^. We shall have
I may first observe that[^] and[^] are not only independent of t
but also of oT. There is only one exception for
k+2=2, 3 or 4.
318
\^
This is due to the fact that, for k + 2 > 4, terms having the
following form
ei'pl ^rjnl ^./a)
which may be included in the second term in one of the equations (21)
can only be independent of t if
since |q| cannot exceed 4 and since qn must be an integer number.
Thus, the second terms of equations (21) are linear and homogeneous
functions of Cq and uq . The coefficients of these linear functions are
absolute constants which are independent of oT.
However, uo must be positive; otherwise Vuo"would be Imaginary. The
equations (21) added to inequality uq > will determine the sign of A.
.- ^rT^f.°^^^ P°^''^ °''^ ^"^^^ ^^^^ ^^8" ^°^^ ^°t depend on u, since equa-
tions (21^ do not depend on it. We have seen that the equation which de-
termines w has two solutions which are actually different
ra = 73(1, w — T3o -I-
A- -T- a
In conformance with the preceding statements, a periodic solution
which will be real if the sign of A is suitably chosen corresponds to each
of them. The choice of this sign does not depend on 15. and these two
solutions will both be real for A > and will both be imaginary for A <
or the opposite will hold true. '
It first appears tha£ two periodic solutions correspond to each solu-
tion of the equation for O), since two systems of values for the unknowns /321
e Co and z\/^ are obtained from the relationships between A, n, z^^^ and
z^. This is not the case, however. Without restricting the conditions
of generality, we may assume that ^/^^^ is positive, because we do not change
our formulas in any way by changing y/^ into -y^. and ^ into (J + ^.
_ Out of our two systems of values, there is only one for which V^ is
positive. ' "
Therefore, we have:
Two real, periodic solutions of the second type for A > (or for
A < 0) .
No solution of the second type for A < (or for A > 0) .
319
.et us again employ the nctatio. given In Chapter XXVIII and. In
particular, that given in No. 331.
U, may be reduced to p^ and corresponds to the tern, containing
x^yi which appears in 0q.
U„ „ay be reduced to a constant factor multiplied hy P^ corresponding
to terms coming fromj^J and |_^^J.
The first term of W which may not be reduced to a power U, has the
following form
pi'-2[Acos(A- -2)-hB]
and comes from Q]f^+2'
The function whose maxima and minima we must study, and which must
play the role of the function
studied on page 247, will have the following form
Ap*+» cos(A: -(- a)? -^ Pp' - ■«?''
„here P Is a whole polynomial In p^ with constant coefficients.
«e have disregarded the particular cases In which the denominator of
n equals 2, 3 or A.
/322
m-scussiop "f Particular Ca_ses
367. Let us assume that this denominator equals 4.
_ rn 1 \^ P^l will no longer be independent of (S, and
Then [02], [^5 J . LduoJ ,,i^
they will include terms containing e
The equation for W will always yield two different solutions
It
-^^i^i z;z^rT^ —rg-ca^e: rayTcc-' -
TWO real solutions of the second type for X > 0; -ro solution for
X < 0;
one real solution of the second type for X > 0; one solution for
320
X < 0;
Zero real solution of the second type for A > Q; two solutions for
A < 0.
The function Uq + zU^ given on page 247 becomes
p'(A cos4f -r- B) — ^p».
Let us now assume that the denominator of n equals 3.
The expansion of \i in powers of e then begins with a term containing
^yuQ, so that if we set y = X, we shall obtain e^^q and ev/uo~ in series
which may be developed in powers of X, and no longer ofyT".
The
sign of yTi^ will depend on m, and if it is positive f or ~ = 53- it
7T
will be negative for u = oig + -r
Therefore, if it is always convenient for us to assume thatyuQ is
mainly positive, we shall readily find that we have:
A real solution of the second type for X > and a real solution of
the second type for X < 0.
The function Uq + zUi given on page 247 becomes
Ap'cos3^ — sp*.
Finally, if the denominator of n equals 2, [©z^shn" . hr~^ > include
terms containing e-'*-'-^, e~^^. /323
The equation for oT takes the form
Acos(4nj-t- B)-f- A'cos(2ra -f-B') =0
and it has eight solutions
•jr 3^
2 a
3i:
rai, CI, H — , cj, -t-TT, nr, H
Of the two terms "aJQ and wj , at least one is real.
The following hypotheses are possible: (4, 0), (3, 1), (2, 2), (1, 3),
(0, 4), (2, 0), (1, 1), (0, 2).
The first number between the parenthesis represents the number of
periodic solutions for X > 0, and the second is the same number for X < 0.
321
The function given on page 247 becomes
Ap»cos i9 H-Bpicos2'f -+- Cp'sin2-+- Dp' — ap'
Application to Equations of No. 13
368. Let us return to the canonical equations of dynamics:
dt " dyC dl dxi
Just as in No. 13, No. 42, No. 125, etc., I shall assume that F is a
periodic function of the y's, which may be developed in powers of a para-
meter y In the following form
F = Fo-H,jiF,-H...
and that Fq depends only on the x's.
We saw in No. 42 that these equations have an infinite number of solu-
tions of the first type
^.• = ?/(0. r/-'f.(0 (2)
where the functions is>^ and ip^ may be developed in increasing powers of y. /324
Let us consider one of these solutions (2) .
Let T be the period, and let a be one of the characteristic exponents.
There will be two of them, which are different from zero, which are equal
and have opposite signs, where we may assume two degrees of freedom.
We saw in Chapter IV that a depends on y, and may be developed in
powers of ^T. When y varies continuously, the same will hold true for
a. For y = yg, let us assume that aT is commensurable with 2iTT and equal
to 2niir.
We may conclude from this that, for y which is close to yQ, there are
solutions of the second type, which are derived from (2) and whose period
is (k + 2)T, where k + 2 designates the denominator of n.
If we put aside the cases in which k + 2 equals 2, 3, or 4, we have
seen that two of these solutions exist when X (here y - yo) has a certain
sign, and that they do not exist when X (here y - yg) has the opposite
sign.
I have stated that the cases in which k + 2 = 2, 3, 4 have been dis-
regarded, and I may do this without causing any inconvenience. The
322
following
aT
= n
2in
may be developed in powers of \/^, and vanishes with yyT For small
values of y, n is therefore very small, and its denominator is definitely
larger than 4.
We therefore have two hypotheses:
Either the solutions of the second type occur only for y > yo, or
they occur for y < yg.
Which of these two hypotheses is valid?
Everything depends on the sign of a certain term Q, which depends
itself on the coefficients of ug and ^o i^
In order to determine this sign, we shall not need to formulate this
term, and the following considerations will suffice.
369. Let us first take a simple case, which will be that presented /325
in No. 199. Let us set
F = Xi -(- t| -H ji COS/j
with the canonical equations
which yields
rfr, _ rfF dy,- _ dF
dt ~ dy'i dl dxi'
§=«. t = -' 'S-"'"'" *-— (1)
The function S of Jacob! may be written
S = xlyt-^ J y/C^ fi cos_^, dy,
2 and C
(2)
with two constants xS and C. We may derive the following from this
J 2/0 — |Jl<
where A and y^ are two new integration constants.
323
\J
It may be seen that the elliptical integral is introduced
J a v/C — li cosjKi
This integral has a real period, which is the integral taken between
and 2." il 1C| > lul, and two times the integral taken between
if Ic! < |y|.
Let us call u this real period.
A periodic solution corresponds to each value of o) which is commensur-
able with 2TT. However, we must distinguish between two cases.
If Id > |y|, yi and ya increase by a multiple of Ztt during one
perio" tL corUsponding periodic solutions are solutions of the fxrst
type.
If Icl < |y|, yi increases by a multiple of 27t during one period. ^326
and yf retims'to it2 original value. The corresponding solutxons are
solutions of the second type.
This discussion must be supplemented by two unusual Periodic solu-
tions which must be regarded as solutions of the fxrst type. Let us set
y > 0, and these solutions will then be
I have stated that it must be assumed that these l^"er solutions are
of the first type, and that the solutions corresponding to |C| |p1 must
be regarded as solutions of the second type.
Let us assign to C a value which is a little higher than -y . and let
us set
C = (e — i)!i,
Where e is very small, yi will not be able to deviate very greatly from ..
We shall approximately have
C — jx cosjKi = H 1 5 '2 J '
and the period o) will be equal to
■It
324
from which we may draw the following conclusions. Let a be an arbitrary
number which is commensurable with 2-w. There is a series of periodic
solutions such that |c| < |y| and that u = a. If |/^is very close to
TT
, C will be very close to -]i, and for
2ct
these periodic solutions will coincide with the second solution (4) which
is of the first type. We may now recognize the characteristic property
of solutions of the second type.
It may be seen that the second solution (4) — i.e., that of the
two solutions (4) which is stable — gives rise to solutions of the second
type, as was explained in Chapter XXVIII.
If the other solutions of the first type — those which are such that
|c| > |y| — do not produce solutions of the second type, this is due to /327
the very particular form of the equations (1). (For these solutions, the
characteristic exponents are always zero.)
Let us first consider solutions of the first type, such that [c| > |y[.
Let us set C = Cq + e. The period w, i.e., the integral (3) taken
between and Ztt, may be developed in powers of e and of y, and the known
terms may be reduced to
Wo'
Let us assign an arbitrary commensurable value to \/^. We shall have
a periodic solution every time that we have
0) = .
/Co
The equation is satisfied for e = y = 0, and we may derive e and,
consequently, C from this equation, in series which develop in powers of
y. The equations (2) will then give us xj and yj developed in powers of
y. These are the expansions of Chapter III.
Let us pass to the second type, such that |c| < |y|. Let us set
C = ey . We shall have
^^J '.
dy\
2 .
Therefore, there is only a periodic solution corresponding to a given
value of 0), which is commensurable with Ztt, if
The coefficients of the Fourier expansion are therefore functions of y,
which are real for
(1) yi
and imaginary for ,- n
VP< — 7=-
to y2
It is apparent that the same line of reasoning would lead to the same
result if, instead of
326
\^
F = xt -h x] -h ij. cos_;'i,
we had set
F=.Fo-+-h[F,],
where Fq depends only on xi and X2, and iFiJ depends only on xj , X2 and /329
yi. The solutions of the second type would still have been real for
y > Pq.
370. In the general case, the quantity Q, which was in question at
the end of No. 368 and whose sign we shall try to determine, obviously de-
pends on y. If y is sufficiently small, the first term of the expansion
will provide its sign.
Let us determine the function S by the Bohlin method, and let us set
S = So+ /jjiSi -+- fjLSi + . ...
If y is small enough, it will obviously be the first two terms
which will be the most important. If we set
F = Fo+,uF,-+-(jlJF, + ...,
we have seen in Chapter XIX that Sq and Sj depend neither on F2 or F^ -
[Fil, but only on Fq and [fJ, where the mean value of Ft is designated
by [Fil.
Let us again take the quantity Q from No. 368. The first term of
its expansion will only depend on Sq and Si, and consequently on Fq and
[Fj]. The same would hold true if we had set
F = F„+,4F,],
which is, consequently, the same as in the preceding section.
In the preceding section we found that solutions of the second type
exist only for
This conclusion still holds in the general case, provided that yg is suffi-
ciently small.
What is the value of yg for which this conclusion would no longer hold?
Let us again consider the notation given in No. 361, which is that of
No. 275. The exponent a which appears there may be developed in powers of
327
\^
the product AA' .
It may be reduced to the characteristic exponent for AA' = 0.
Since we assume that the solution of the first type is stable ^^^^330
a is imaginary. A and A' are imaginary and conjugate, and the product AA
is positive.
For small values of y, a decreases when AA' increases. If the re-
verse were true, solutions of the second type would exist only for
The desired value of Uo ±^ therefore that for which a ^^f ^^^°^^- _
crease when AA' increases. It is therefore that which cancels the deriva
tive of a with respect to AA' .
328
CHAPTER XXXI
PROPERTIES OF SOLUTIONS OF THE SECOND TYPE
Solutions of the Second Type and the Principle of Least Action /331
371, I cannot pass over the relationships between the theory of solu-
tions of the second type and the principle of least action in silence. I
wrote Chapter XXIX just for these relationships. However, in order to
understand them some preliminary remarks are still necessary.
Let us assume two degrees of freedom. Let x^ and X2 be the two
variables of the first series, which may be regarded as the coordinates
of a point in a plane. The plane curves which satisfy our differential
equations will comprise what I have designated as trajectories .
Let M be an arbitrary point in the plane. Let us consider the group
of trajectories emanating from the point M, and let E be their envelope.
Let F be the niS- kinetic focus of M on the trajectory (T) . This trajec-
tory will touch the envelope E at the point F, according to the definition
of kinetic focuses. I would like to recall that the nt£ focus of M, or its
focus of the order n, is the n^ point of intersection of T with the in-
finitely adjacent trajectory passing through M. However, the conditions
of this contact may vary. It may happen that F is not a singular point of
the curve E, and that the contact is of the first order. This is the most
general case.
Let
a-i = 0(2-1)
be the equations of the trajectory (T) and of a trajectory (T' ) which is
very close, emanating from the point M.
Let zi and Z2 be the coordinates of the point M, and let ui and U2 /332
be the coordinates of F. Since (T) passes through M and F, and since (T' )
passes through M, we shall have
Due to the fact that the trajectory (T') is very close to (T) , the
function if) will be very small. I may call a the angle at which two tra-
jectories Intersect the point M. It is this angle which will define the
trajectory (T'), and the function ii will depend on the angle a. It will
be very small if, as we have assumed, this angle ct is itself very small,
and it will vanish with a.
The value of i;;'(z2) (designating the derivative of i|j by i/;') will have
329
\^
sign if F is a focus of odd order.
one characteristic of the case in which we -^/^^-^^f ,f,,^'^ '"^'
that Hu2) is of the same order as a^. and always of the same sxgn.
For example, let us assume that ^Cuz) is positive.
If the sign of a is such that ^' (uz) is positive, the trajectory
(T') w 1 inteSect (T) at a point F' which is close to the PO- F and
-ir^ -^LferE^rfo^rF^fShiL^anrchirF ^^^i^: ^^.
^ra'w nU::; li^e'of reasoni;g. the action is larger (at least .n abso-
lute motion) when we pass from M to F' proceeding along (T ) than xt
when we pass from M to F' proceeding along (T) .
If the sign of a is such that *' (u^) is negative. (T) intersects (T)
. -nrv' which is farther away from M than F. In this case, (T )
? u^h^faftef F-! i^dlx) touche ^^^^f •'' ' lof/ y^ '^'^ ^"^ " ^
F', the action is greater along (T) than it is along (T ).
The results would be just the opposite if ^(u,) -f ^^^^ff J^^;^ l^'
sect (T) close to F and just short of F.
In this case, we may say that F is an ordinary focus .
It cannot happen that F is an ordinary point of E, and that the con-
tact is of a higher order than the first.
Let us develop ij;(x^) in powers of a. and let us set /333
The condition under which there would be a contact of higher order
But we already have
i;>,(«,) = o
^e coefficient of the second derivative is reduced to unity.
330
If the integral tf^i (X2) vanishes, as well as its first derivative,
for X2 = U2, it would be identically zero, which is absurd.
Therefore, there is never a contact of higher order.
However, it may happen that F is a cusp of the curve E. Either the
cusp point is on the side of M, so that a moving point proceeding from M
to F will encounter M with the cusp point directed at M, or the cusp point
is turned the opposite way so that the moving point encounters M with the
cusp point turned away from M. In the first case, I shall state that F
is a pointed focus , and in the second case I shall state that F is a taloned
focus .
In one and the other case, 4'(u2) is on the order of a^. In this case,
the pointed focus has the sign of a, if P is a focus of odd order, and it
has the opposite sign of a if F is a focus of even order. The opposite is
true iu the case of a taloned focus.
In the case of a pointed focus, all the trajectories (T') intersect
(T) at a point F' which is close to F and beyond F. Proceeding from M
to F' , the action is greater along (T) than it is along (T').
In the case of a taloned focus, all the trajectories (T') intersect
(T) at a point F' which is close to F and just short of F. Proceeding
from M to F' , the action is greater along (T') than it is along (T) .
Let F' be a point of (T) which is sufficiently close to F. In the
case of a pointed focus, I may join M with F' by a trajectory (T'), if F'
is beyond F. In the case of a taloned focus, I may join M with F' if F'
is just short of F.
It could finally be the case that F is a singular point of E which is /334
more complicated than an ordinary cusp. I would then state that it is a
singular focus .
I would only like to note that we cannot pass from a pointed focus
to a taloned focus except through a singular focus, because at the time of
passage iij(u2) must be of the order a"*.
372. Let us now consider an arbitrary periodic solution. It will
correspond to a closed trajectory (T) . Let a be the characteristic ex-
ponent and T be the period. In Chapter XXIX we saw how to determine
successive kinetic focuses (No. 347).
Let us assume that a equals — = — , where n is a commensurable number
whose numerator is p. In this case, the application of the rule given
in No. 347 shows that each point of (T) coincides with its 2p^ focus.
331
If, just as in No. 347, we take a unit of time such that the period
T equals Ztt, we have a = in. If we designate the value of the function
T at the point M by tq, and if xi, T2, ..., T2p are the values of this
function T at the first, second up to the 2p^ focus of M, according
to the rule given in No. 347, we shall have the following
"0 = ) • ■ •) ij/j -0 -
a n
If p is the numerator of n, it can be seen that Xjp - tq is a multi-
ple of 2tt, i.e., that M and its 2p^ focus coincide.
The trajectory emanating from the point M which is infinitely close
to (T) will therefore pass through the point M again after having gone
around the closed trajectory (T) k + 2 times, if k + 2 is the denominator
of n.
The point M is therefore its 2p^ focus. However, we may wish to
know what category of focuses it belongs to, from the point of view of the
classification presented in the preceding section.
Let us adopt a system of coordinates which are similar to the polar
coordinates, so that the equation for the closed trajectory (T) is
and so that w varies from to 2tt when one passes around this closed tra-
jectory. The curves p = const, are then closed curves which form an en- /335
velope around each other in the same way as concentric circles. The
curves cj = const, form a bundle of divergent curves which intersect all
the curves p = const., in such a way that the curve u = a + 2it coincides
with the curve a> = a.
Then let coq be the value of w which corresponds to the point of de-
parture M. The value of w which will correspond to this same point M,
regarded as the 2p^ focus of the point of departure, will be
Let
0|)-t- 2( A- -i- 2)7t.
p — 1 -r ({/(w)
be the equation of a trajectory (T') which is close to (T) and passes
through M. The function i|j(u)) will correspond to the function ((j(x2) given
in the preceding section. We shall have )|;(a)o) =0, and we must now discuss
the sign of
332
We must therefore formulate the function !l;(a)), and for this purpose
we need only apply the principles of Chapter VII, or the principles given
m No. 274. For example, if we apply the latter principles, we shall ob-
tain the following. The function i/j(a)) may be developed in powers of the
two quantities
The coefficients of the expansion are periodic functions of the
period 27t; A and A' are two integration constants. With respect to a,
it is a constant which may be developed in powers of the product AA'
I -^ a„ -;- a,(AA') -,- cc,(A,Vp -i-
The term qq equals the characteristic exponent of (T) , i.e. it
equals in. '
If (T') differs very little from (T) , the two constants A and A' are
very small. They are on the order of the angle which I called a in the /336
preceding section, and which must not be confused with the exponent which
I have designated by the same symbol in the present section.
If we take the approximation up to the third order inclusively with
respect to A and A', ^(w) will be reduced to a polynomial of the third
order with respect to these two constants, and I may then write
■{-(w) = Ae'wcr-t- A'e-!'w) = Acre'>>f».+a,AA,^.^'j'^_u(3.*j,AA'i_j_y(Agoi.o), A'e-".'-')
or
-)-st,(oAA'(Ae«.'Oj- A'e-='oWa') + /(Ae»o« A'e-».").
When 0) increases by (2k + 4)^, the coefficients of f , as well as
o and a', do not change. The same holds true for e"0'^, since — = n has
k + 2 for the denominator. Therefore, the same still holds true for
Ae'."a, A'e-".'",', /(Ae=<.^) = (A- + 2)^^. AV(A e-."a - AV-^.-O-
However, ^(wq) is zero. The term whose sign we must determine is
therefore
(2/.- + 2):ti, AA'(Ae»='"o!7j_ A'e-»«".!j;, ).
I Shall employ Oq and c/q to designate the values of a and o' for o) = (.q •
I should first point out that this term is of the third order which, i337
according to the preceding section, indicates to us that our focuses wxll
m general be pointed focuses or taloned focuses. It may now be stated
that this term always has the same sign, and that its coefficient cannot
vanish.
The two constants A and A' are related by the following relationship
t;/( 4, we will only be finished with the calculation of the terms of
higher order, which are not included in the preceding analysis.
374. For example, let us assume that all the focuses are pointed.
Let M be an arbitrary point of (T) ; this point will be the 2p^ focus
with respect to itself. Let M' be a point located a little beyond the
point M in the direction in which the trajectory (T) and the trajectories
close to (T') are traversed. I may draw a trajectory (T') emanating from
point (M) , which will deviate very little from (T) , which will pass around
(T) k + 2 times, which will finally end at the point M' , and which will
have 2p + 1 points of intersection with (T) , counting the intersection
points M and M' .
Due to the fact that the focus is a pointed focus, the trajectories
(T') which are close to (T) will all intersect (T) again beyond the focus.
We may therefore draw the trajectory (T*) which satisfies the conditions
I have just discussed, provided that the distance MM' is smaller than 6.
It is apparent that the upper limit, which must not exceed the distance MM',
depends upon the position of M on (T) . However, it never vanishes, since
there is not a singular focus. It is therefore sufficient for me to set
6 equal to the smallest value which this upper limit can take on, and I
shall assume that 5 is a constant.
Therefore, if the distance MM' is smaller than 6, we may draw a tra-
jectory (T') satisfying our conditions. We may even draw two of them, one
intersecting (T) at M at a positive angle, and the other intersecting it
at a negative angle.
Under this assumption, let us assume that our differential canonical
equations depend on the parameter X. For A = 0, the closed trajectory (T)
has ag = in as the characteristic exponent. Let us assume that, for X > 0,
the characteristic exponent divided by i is larger than n, and that for
X < 0, on the other hand, it is smaller than n.
For X ^ 0, the point M will no longer be its own 2p£ll focus. Its
2p^ focus will be located a little short of M for X > 0, and beyond M
335
K^
-' X < 0. -t . .e th- focus X.e distance » -n_^na.„aU. .epen.^^^
Slf ^is": e!" ? 13°; a ;„t\haf c 'llAe a co.tlnuou. function of
H il i: rAl-^foJt :; aSaJ-f ?f?f -r^ r J.^-ac^oo^/fA.
to thrrnncljlei given L No. 347. depending on the value of the charac-
teristic exponent. The distance MF can never vanish.
Let F- be a point located a little beyond F. We may connect M with
F' bv a traiectory (T'), provided that the distance FF' Is less than a
certlin J^^titH'. It L apparent that 6' is a continuous function of
X, and that it may be reduced to 6 for X - 0.
Let us set X > 0, In such a way that M is beyond F. We may have M
play ihe "le of F' . and we may connect M to itself by a trajectory (T ),
provided that the distance MF is smaller than 6 ' , or provided that
e<8'
For X = 0, e is zero, and 6 ' =6 > 0. Therefore, we may take X small
enough so that'the inequality is satisfied.
we may then connect the point M to itself through a trajectory (T')
deviating a little from (+) , passing around (T) k + 2 txmes, and inter-
secting (T) 2p + 1 times.
Figure 12
In the figure, BA represents an arc of (T) on which M is located.
MC is L arc ofcT') starting from M and DM is another arc of this same
trajec'rj bordering upon M. The arrows indicate the direction xn which
the trajectories are traversed.
The noint M may also be connected to itself not through one trajec-
tory.^ut' through So (T'). For one, as the figure i-^i-tes the angle ^340
^"^Is positive! so that CM is above MA. For the other, the angle CMA
336
. would be negative.
The trajectory (T') must not be regarded as a closed trajectory. It
leaves the point M to return to the point M, but the direction of the
tangent is not the same at the point of departure as it is at the point
of arrival, so that the arcs MC and DM do not join each other.
The trajectory (T'), thus proceeding from M to M with a hooked angle
at M, will form what may be called a loop . If the same construction is
followed for the points M of (T) , we shall obtain a series of loops . We
shall obtain two of them, the first corresponding to the case in which the
angle CMA is positive, and the second corresponding to the case in which
this angle is negative. These two series are separated from each other,
and the passage from one to another may only be made if the angle CMA is
infinitely small.
The trajectory (T'), which is infinitely close to (T) , would pass
through the focus F, according to the definition of focuses. However,
since it must end at the point M, the points M and F 5^ould coincide, and
this cannot happen according to the principles presented in No. 347.
Therefore, if all of the focuses are pointed, we have two series of
loops for A > 0, and we have no more for A < 0.
If all the focuses were taloned, the same line of reasoning could be
repeated. We would find that there are two series of loops for X < 0, and
that there are no more for X > 0.
375. Let us consider one of the series of loops defined in the pre-
ceding section. The action calculated along one of these loops will vary
with the position of the point M; it will have at least one maximum or
one minimum.
If the action is maximum or minimum, it may be stated that the two
arcs MC and CD coincide, so that the trajectory (T') is closed and corres-
ponds to a periodic solution of the second type.
For example, let us assume that the trajectory (T') corresponds to
the minimum of the action, and that the angle CMA is larger than the angle
BMD, just as in the figure. Let us then take a point Mi to the left of
M and infinitely close to M, and let us construct a loop (T?) which /341
differs by an infinitely small amount from the loop (T'), having its
hooked point at Mi. Let MiCi and MiDi be two arcs of this loop.
From M and from Mi I may draw two normals MP and MiQ on MiCi and MD.
According to a well-known theorem, the action along (T') from the
point M up to the point Q will equal the action along (Tj) from the point
337
\^
P to Ml. We shall therefore have
action (T',)== action (T') -i- action (.M,P) — action (MQ)
or
action (T'j)=: action (T') -4- action (.\lM,)(cosCMA — cosBMQ),
or finally
action (T'j )< action (T'J,
which is absurd, since (T') was assumed to correspond to the minimum of
the action.
If we set
CM A < B.MD,
we would arrive at the same absurd result placing Mi to the right of M.
We must therefore assume that
CM A = BMD,
i.e., that the two arcs coincide.
The same line of reasoning may be applied to the case of the maximum.
Each series of loops therefore contains at least two closed trajec-
tories.
Each of these closed trajectories passes around (T) k + 2 times,
and intersects (T) at 2p points. For p of these points, the angle similar
to CMA is positive, and for the other p points, it is negative. Due to the
fact that the curve (T') is closed, it must intersect (T) as many times m
one direction as in the other direction.
Therefore, it may be assumed that this closed trajectory consists of
2p types of loops . because we may regard any arbitrary one of our 2p points
of intersection as the hooked point. For p of these types, the loop thus
defined would belong to the first series, and for the other p types, it
would belong to the second series.
Among the loops of each series, there are therefore not two, but at
least 2p of them, which may be reduced to closed trajectories. However, JJ^
one thus obtains not 4p , but only two different closed trajectories.
The fact that there are not more of them is, in general, not the
result of the preceding line of reasoning, but may be concluded from the
principles presented in the preceding chapter.
338
The trajectory (T') thus defined will have yCk + l)p double points,
if k is odd, and — (k + 2)p double points if k is even. This is valid for
small values of A, and it remains valid no matter how large \ may be as
long as (T') exists. The number of double points could only vary if two
branches of the curve (T') were tangent to each, other. However, two tra-
jectories cannot be tangent to each other without coinciding.
For the same reason, no matter how large X may be, as long as the two
trajectories (T) and (T') exist, they will intersect at 2p points.
376. The entire line of reasoning presented In the preceding section
assumes that we are dealing with absolute motion .
If this line of reasoning is extended to the case of relative motion,
difficulties will be encountered which are not insurmountable, but which I
shall not try to surmount at this point.
To begin with, we must modify the construction employed in the preceding
section. Instead of drawing MP and M^Q normal to MjCi and MD, we must pro-
ceed as follows. In order to construct MP, for example, we should construct
a circle which is infinitely small and which satisfies the following condi-
tions. It intersects M^Ci at P and touches the line MP at this point. The
line connecting M to the center must have a given direction, and the ratio
of the line length to the radius must be given. The line MP thus constructed
has the same properties as the normal in absolute motion. Unfortunately, in
certain cases this construction entails certain difficulties.
In addition, the action (MM^) is not always positive. If it became zero,
this line of reasoning would still have a defect. The maximum or the minimum
could be reached at the point M, so that the action (MMi) is zero, and this
could occur without the necessity of the arcs MC and DM coinciding. /343
Our line of reasoning therefore only applies to the case of relative
motion, if the action is positive along (T) .
In any case, one of the conclusions is still valid. The closed trajec-
tory (T') always exists, since — if the line of reasoning given in the pre-
ceding section is lacking — the same does not hold true for the line of
reasoning given in Chapters XXVIII and XXX. In addition, (T') intersects
(T) at 2p points, and has ^(k + 1) or |-(.k + 2) double points.
This is valid for small values of X, but it cannot be concluded any
longer that this is valid no matter what A may be, because two trajectories
may be tangent without coinciding, provided that they are traversed in the
opposite direction.
339
stability and Instability
377. Let us assume that there are only two degrees of freedom, two
of the characteristic exponents are zero, and the two others are equal and
have opposite signs.
The equation which has the following as roots
is an equation of the second order whose coefficients are real (T represents
the period and a represents one of the characteristic exponents) .
Its roots are therefore real or imaginary and conjugate.
If they are real and positive, the ot's are real, and the periodic solu-
tion is unstable.
If they are imaginary, the a's are imaginary and conjugate. Since the
product equals +1, the a's are purely imaginary, and the periodic solution
is stable.
If they are real and negative, the a's are imaginary but complex , with
the imaginary part equalling y". The periodic solution is still unstable.
They cannot be real and have opposite signs, since the product equals
+1.
There are therefore two kinds of unstable solutions, corresponding to
the following two hypotheses /344
The passage from stable solutions to unstable solutions of the first
type occurs for the value
The passage from stable solutions to unstable solutions of the second
type occurs for the value
"= T-
378. Let us first study the passage to unstable solutions of the first
type. At the moment of passage, we have
340
Let us again consider the terms gj^ and i)^ defined in Chapter III, and
let us consider the equation
d?,
d?,
d?l
d^i
dfr
dft
d'W
dh
o-O/,
d^,
dh
dh
rf'l.
d^.
dh
dh
d^. ^
dh ^
dh
d^,
dh
dh
CD
This equation has the following roots
o, o, e«T__[^ e-iT_,.
At the time of passage, the four roots become zero.
Before studying the simple case in which we are dealing with equations
of dynamics with two degrees of freedom, and in which we assume that the
function F does not depend explicitly on time and that, consequently, the
equations have the energy integral F = const., it is advantageous to consider
for a moment a case which is even simpler.
Let F be an arbitrary function of x, y and t, which is periodic of
period T with respect to t. Let us consider the canonical equations
7345
dx
dJ
dF
dt
dF
dx''
(2)
These are the equations of dynamics with only one degree of freedom. How-
ever, due to the fact that F depends on t, they do not have the energy equa-
tion F = const.
Let us assume that these equations (2) have a periodic solution of
period T. The characteristic exponents will be provided by the following
equation which is similar to (1)
<^1 «
dh~
dh dh ~
which has the following roots
hi
(3)
e'T_,
These roots all become zero at the moment of passage.
Let us assume that F depends on a certain parameter y and that, for
X = 0, the two roots of the equation (3) are zero. The functions ipi and ^pz
will depend not only on Bj and 62. but also on y. We shall assume that F
341
may be developed in powers of y, and that consequently ^ and ^2 may be
developed in powers of 61, B2 and y.
The periodic solutions will be provided by the following equations
(4)
ij^j =0.
For y = 0, Pi = B2 = 0, the functional determinant of the i|;'s with^_
respect to the P's is zero. However, in general the four derivatives ^
will not vanish at the same time. For example, let us assume
and we shall derive Bi in series developed in powers of B2 and y from the
?irst equation (4), aid we shall substitute it in the second equation (1). i346
Let
be the result of the substitution. Our functional determinant being zero,
we shall have
However, we may distinguish between two cases:
1. The derivative j^ is not zero, or, in other words, the functional
determinant of i>^ and ^^ with respect to B^ and y is not zero.
In this case if we assume that B2 and y are the coordinates of a point
in a pLne' the c^rve represented by equation (5) will have an ordinary
point at the origin, where the tangent will be the line y - 0.
In general, the second derivative
Will not be zero, i.e.. the origin will not be a point of inflection of the
curve (5) .
If we intersect the line y = yo, where yo is a rather small constant,
we may have, two points of intersection for this line and the curve (5) in
the vicinity of the origin, or we may not have any, depending on the sign
of yg.
342
For example, if this curve is above its tangent, we shall have two
intersections for uq > 0, and consequently two periodic solutions, and
for Mo ^ we shall not have any.
We have thus seen two periodic solutions approach each other, coincide,
and then disappear.
Let us consider the two points of intersection of the line y = yg
with the curve (5) . They will correspond to two consecutive roots of the
equation (5) and, consequently, to two values having opposite signs of the
derivative ■^^, and therefore to two values of opposite signs of the func-
tional determinant of the ifi^'s with respect to the B's, that is, of the /347
product
i.e. , of a^ .
Therefore, one of the two periodic solutions which coincides then to
disappear is always stable, and the other is unstable .
2. The derivative ^ = 0, or in other words the functional determinant
of ^i and 1^2 with respect to 6i and y, is zero.
The curve (5) then has a singular point at the origin which, in general,
will be an ordinary, double point.
Two branches of the curve intersect at the origin, and the line y = Uq
will always meet the curve at two points. We shall therefore have two
periodic solutions, no matter what the sign of \iq may be.
The two branches of the curve determine four regions in the vicinty
of the origin. In two of these regions which are opposite the peak ¥ will
be positive; in the other two regions, it will be negative.
Let OPi , OP2, OP 3, OPi+ be the four half-branches which converge at the
origin. OPi will be the extension of OP3 and OP2 will be the extension of
OP^. OPi and OP2 will correspond to yg > 0; OP3 and OP4 will correspond to
yo < 0. The function "V will be positive for the angles P1OP2, P3OP1+, and
negative for the angles P2OP3, P1OP1+.
We have just seen that the stability depends on the sign of the deriva-
df
^ diT* ^°^ example, when we pass over OP2 , 4* will change from negative
to positive. The derivative will be positive, and the solution will be
stable, for example. It will also be stable when we pass over OPi^, and
unstable when we pass over OP2 or OP3.
343
The periodic solutions corresponding to OPi are stable and they form
an analytical sequence with respect to those which correspond to OP3 and
which are unstable.
Conversely, those which correspond to OP2 and which are unstable are
the analytical sequence with respect to those which correspond to OP,, and
which are stable.
We thus have two analytical series of periodic solutions which coincide
for y = 0, and at this instant of tiir.e the two series excha nge tneir stabil -
We have just studied the two simplest cases, but there may be ^ multi-
tude of other cases corresponding to different singularities which the IJh^
curve (5) may have at the origin.
However, no matter what these singularities may be, we shall observe
an even p + q number of half-branches emanating from the origin i.e. p
forTT and q for u < 0. Let us assume that a small circle about the
origin encounters them in the following order
OP,, OP, 0?,,+^.
Let
OP,, 0I'„ ..., OP,, (6)
be those which correspond to y > and let
0P;,-„ or,,^„ ..., ov,^,
be those which correspond to y < 0.
Then the half-branches (6) will correspond alternately to periodic
stable solutions and to unstable solutions. For purposes of brevity, I
may state that these half-branches are alternately stable or unstable.
The same holds true for the half-branches (7).
In addition, OPp and OPp^.^ are both stable or both unstable.
Consequently, the same holds true for OPp+q and OP^.
Therefore, let p' and p" be the number of stable half-branches and the
number of unstable half-branches for y > 0, so that we have
p' + p" = p-
Let q' and q" be the corresponding numbers for y < 0, so that q' + q" = q-
There are therefore only three possible hypotheses
344
P' = p' :-', 7'= ?'-'-'.
p' = p"-u q'=q'—\-
In any case, we have
Let us assume that p does not equal q, and, for example, that p > q, /349
in such a way that a certain number of periodic solutions disappears when
we pass from p > to y < 0. It may be seen that this number is always
even, and in addition as many stable solutions as unstable solutions would
always disappear , according to the preceding equation.
Let us now assume that wq have an analytical series of periodic solu-
tions and that, for u = 0, we pass from stability to instability, or vice-
versa (in such a way that the exponent a vanishes). Then q' and p" (for
example) are at least equal to 1. Therefore, p' + q" is at least equal to
2. It follows from this that we shall have at least another analytical
series of real, periodic solutions which intersect the first for y = 0.
Therefore, if. for a certain value of y. a periodic solution loses sta-
bility or acquires it (in such a way that the exponent a is zero) it will
coincide with anoth er periodic solution , with which it will have exchanged
its stability.
379. Let us now return to the case which I was first going to discuss
— that in which the time does not enter explicitly in the equations, where,
consequently, we have the energy integral F = C, where finally there are
two degrees of freedom.
I shall pursue the same line of reasoning as was the case in No. 317,
and I shall assume that the period of the periodic solution, which Is T for
the solution which corresponds to y = 0, B^ = 0, equals T + t, and differs
very little from T for adjacent periodic solutions. I shall write the
following equations
(1)
A, = 0, -}', = o, -Ij^o, F = Go,
which include the following variables
P'' h, P), Pi, ^, T.
According to our hypotheses, the functional determinant of the i|;'s
with respect to the B's must vanish, as well as all its minors of the first
order. However, the minors of the second order will not all be zero at the
same time, in general.
Therefore let us set 6i = in equations (1), and let us consider the
345
K^
functional determinant A of /35Q
•h. ^" '>» F
with respect to p„ p,, Pv, t.
This determinant vanishes when the g's, y's and x's vanish, but in
general the minors of the first order will not vanish.
Let us consider the functional determinants of F and of two of the
four functions i> with respect to x, and with respect to two of the four
variables 6. Can they all be zero at the same time?
According to the theory of determinants this could only happen if
the following were true:
1. All the minors of the two first orders of the determinants of the
IP's with respect to the x's were zero at the same time, which does not
occur, in general, and which we shall not assume.
2. The derivatives of F were all zero at the same time. We saw in
No. 64* that they must be zero all along the periodic solution. We shall
no longer assume this.
3. The derivatives of the i|j's and of F with respect to x were all zero
at the same time. The following values
would not correspond to a periodic solution strictly speaking, but to a
position of equilibrium (see No. 68).
We shall no longer assvraie this .
We may therefore always assume that all the minors of the first order
of A are not zero.
Let us then eliminate four of our unknowns B and x among the equations
(1).
For example, let us eliminate 6i, B3, 04, ^ ; we shall still have an
equation of the following form
Due to the fact that this equation has exactly the same form as equation (5)
of the preceding section, it will be handled in the same way, and we shall
arrive at the same results:
346
1. When periodic solutions disappear after having coincided, an even
number, and as many stable as unstable solutions, always disappear.
2. When a periodic solution loses or acquires stability when we vary
VI continuously (in such a way that a vanishes), we may always be certain /351
that at the moment of passage another real, periodic solution of the same
period coincides with it.
380. Let us proceed to the second case, that in which
Due to the fact that none of the characteristic exponents vanishes for
H =0,
except the two which are always zero, there is no periodic solution of period
T which coincides with the first for
fx = o.
On the other hand, according to principles presented in Chapter XXVIII,
there are periodic solutions of the second type, of period 2T, which coin-
cide with the given solution whose period is T for y = 0.
What may we say regarding their stability? For y > 0, for example, we
shall have a stable solution of period T which will become unstable for y < 0.
For y > 0, let p' and p" be the number of stable solutions and the num-
ber of unstable solutions which have the period 2T, without having the period
T. Let q' and q" be the corresponding numbers for y < 0.
Let us then consider all the solutions of period 2T, whether they have
have the period T or not. Applying the principles presented in No. 378 to
them, I find that I may postulate the following three hypotheses regarding
these four numbers :
7.+p' = p\ q=q'.
However, if we refer to the principles given in Chapter XXVIII, we
shall find that these four numbers cannot take all values which are compatible
with the three hypotheses. The simplest and most frequent cases are investi-
gated in No. 335.
347
Application to the Orbits of Darwin i}^
381. In Volume XXI of Acta Mathematica , M. G. H. Darwin studied cer-
tain periodic solutions in detail. He discusses the hypotheses given in
No. 9, and considers a perturbing planet which he calls Jupiter, and to
which he attributes a mass which is ten times smaller than that of the Sun.
This fictitious planet describes a circular orbit around the Sun, and a
small perturbed planet having zero mass moves in the plane of this orbit.
He has thus acknowledged the existence of certain periodic solutions
which are again Included in those which I have called solutions of the
first type, and which he has studied in detail. These orbits are referred
to moving axes, turning around the Sun with the same angular velocity as
Jupiter. These orbits are closed curves, in relative motion with respect
to these moving axes.
M. Darwin has called the first class of periodic orbits the class of
planets A. The orbit is a closed curve encircling the Sun, but not encircling
Jupiter. The orbit is stable when the Jacob! constant is larger than 39, and
unstable in the opposite case. The instability corresponds to a characteris-
iiT
tic exponent having — as the imaginary part .
For values of the Jacobi constant which are close to 39, there are
therefore periodic solutions of the second type whose period is double.
The corresponding orbit will be a closed curve with a double point
passing around the Sun twice. The two loops of this curve differ very little
from each other, and both differ very little from a circle.
We shall study these solutions of the second type in greater detail at
a later point.
M. Darwin also obtained oscillating satellites which he called a and b,
and are those which we discussed in No. 52. They are always unstable.
Finally, he obtained satellites which, strictly speaking, with respect
to the system of moving axes under consideration, describe closed curves en-
circling Jupiter, but not encircling the Sun. /353
For C = 40 (C is the Jacobi constant) , we have only one satellite A
which is stable. For C = 39.5, the satellite A becomes unstable with a real
exponent a. However, we have two new satellites B and C, the second of
which is stable, and the first of which is unstable with a real exponent a.
For C = 39. we obtain the same result. For C = 38.5, the satellite C becomes
iiT
unstable with a complex exponent a (whose imaginary part is — ). Finally,
for C = 38, we obtain the same result.
348
We must therefore consider three passages:
1. The passage of satellite A from stability to instability;
2. The appearance of the satellites B and C;
3. The passage of satellite C from stability to instability;
The last two passages do not entail any difficulties.
Two periodic solutions B and C will appear simultaneously which differ
very little from each other. One is stable and the other is unstable; the
exponent a is real for the unstable solution. This conforms with the con-
clusions reached in No. 378.
The passage of the satellite C from stability to instability no longer
presents any difficulties, because the exponent a is complex in the case of
mstabxlity. The conditions presented in No. 380 therefore hold. We there-
fore have periodic solutions of the second type corresponding to closed
curves which pass around Jupiter twice.
382. On the other hand, the passage of satellite A from stability to
instability entails great difficulty, because the exponent a is real in the
case of instability. According to No. 378, we should therefore have exchange
of stability , with other periodic solutions corresponding to closed curves
passing around Jupiter only once. This would not seem to result from the
calculations of Darwin.
We are naturally led to think that the unstable satellites A discovered
by Darwin do not represent the analytical extension of the stable satellites
A
A
Other considerations lead to the same result.
The stable satellites A have ordinary closed curves for orbits; the /354
unstable satellites A have orbits in the form of a figure eight.
How may we pass from one case to another? This may only be done by a
curve having a cusp, but the velocity must be zero at the cusp and. for
reasons of symmetry, this cusp could only be located on the axis of the x's
It could not be between the Sun and Jupiter. In Figure 1, Darwin gives the
curves of zero velocity. For C > 40, 18, these curves intersect the axis
of the X s between the Sun and Jupiter, but this no . longer holds for C < 40
18, and the passage occurs between C = 40 and C = 39.5. '
We are left with the hypothesis that the cusp is located beyond Jupiter
but this is no longer satisfactory. Let us compare the two orbits corres-
ponding to C = 40 and to C = 39.5. The first Intersects the axis of the x's
349
*- . ■.--,• «hi- an^le once beyond Jupiter and once just short of it.
Stl ndVbe t^: -o'inte'LecIion poLts. In the sa.e way the second
nlLt Hf we disregard the double point) intersects the axxs of the x s
r^rl It a rieht Se once beyond Jupiter, and once just short of Jupxter.
Ziyj ? bl Se t^o intersection points. Let us consider the inter-
section point P or P' which is beyond Jupiter, and let us determine the
sigt. of ^. We shall see that this sign is positive for one orbit or the
other. However, ^ would have to change sign when passing through the
cusp.
The point P, the hypothetical cusp, and the point P' cannot therefore
h. rPBarded as the analytical extension of each other. We must then
assumption.
Therefore, I may conclude that the unstable satellites A are not the
analyticaiextension of the stable satellites A. But when do the satel-
lites A become stable?
/355
I can only formulate hypotheses on this point and, in order to do
.u ■ T ?t would be necessary to reconsider the mechanical quadratures
f S'^ia^^ii Soiever ?f we^amine the behavior of the curves, it appears
that'a?Tc:;ta!n time 'the orbit of the satellite A must pass through Jupxter,
Ind that it then becomes what M. Darwin has called an oscilMtxn^,satellxte.
383. Let us study the planets A in greater detail, and the passage of
these planets from stability to instability.
ThP orbits of these planets correspond to what we have designated as
^...^: ^olu^L^s ofthe^irst type (No. 40^. The orbit with a oubepoxnt,
w hich passes around the S un twice and which differs very Ixttle from that ot
Ihe planet 1 at the moment when the orbit of this planet has Just become
unstable! corresponds to which we have designated as periodxc solutxons of
the second type (47) .
If we apply the procedure by which we deduced periodic solutions of the
second t'eTrom those of the first type to solutions of the fxrst type, we
shall obtain solutions of the second type exactly.
Tr. eoi„rinn<, of the second type, the mean anomalistic motions, which
differ verj little from the mean motions strictly speaking, are in a co:™nensur-
tble rali^ We must therefore consider the case in which, for our solution
of the second type (and. consequently, for the planet A at the time of
350
passing from stability to unstability) , the ratio of the mean motions is
close to a simple commensurable number. Since the orbit must pass around
the Sun twice, this ratio will be close to a multiple of i.
2
In other words, at the moment of passage, the term which M. Darwin
has called nT must be close to a multiple of it.
In effect, this is what occurs. The tables of M, Darwin provide us
with the following
C = ,{o A stable, nT -~ ii>4°,
C=-^39,5 A stable, «T = i6>°,
C " 3r) AunstablejrtT = 177",
G = 38,5 Aunstablej,/iT = 191°.
It can be seen that the passage must be made around nT = 170°, and /356
this number is close to 180°.
The mean motion of the planet A is therefore almost three times that
of Jupiter.
We could consider applying the principles presented in Chapter XXX
to a study of these solutions of the second type, but several difficulties
would be encountered because we would be dealing with an exception. It
would be better to resume this study directly.
384. Let us again consider the notation given in No. 313, and let us
set the following, just as in this section
j-i =- L - G, 3-, = L + G,
F' = R-i-G.^ Fo+pF,-f-...,
The term L must have the same sign as G (see page 201, in fine), and the
eccentricity must be very small. Since xi is on the order of the square
of the eccentricity, this variable will also be very small.
Since we only wish to determine the number of periodic solutions and
their stability, we shall be content with an approximation.
We shall therefore neglect y^p^ and the following terms. In the term
yFi , we only take into account secular terms and terms with a very long
period, and we shall neglect the powers which are higher than x^. We
shall have
351
ri^-a-^hx,-h cxi
cosuj,
Where a. b, c are functions only of .„ and where cxicos. is the very long
period term which has been retained.
The very long period terms are terms with I + 3g - 3t, i.e., terms
with 2y2 - yi- We therefore have
lo = iyi — iyi-
We then have
and we may apply the method of Delaunay.
/ 357
The canonical equations have the integral
Xt -+- 2x1 = k,
from which we have
p., ? , ^ _ Ifil _H ,x{a i-6x,-HCJr,cos(d).
With the approximation which has been adopted, we may replace a, b, c by
aa — 2X,a'„, bo, Co,
designating that which a, fj, h, c, become by ao , a'„. b„. c„ when we re-
place X2 by k. Thus,
designate the constants which depend on k, and we have
p^ _ . ^ >. 1— iJ^ -t- ijL(a -^ ^Xi -4- YX, cosoj).
(A- - - x, ;' ■ 2 -2
Let us assume that k is a constant, v'^cos'% v/^sln'^ are rectangular
coordinates of a point in a plane, and let us compile the curve
F' = G,
where C designates a second constant.
This curve also depends on the two constants k and C. If it has a
doubirjoinrthis double point will correspond to a periodic solutxon.
352
which will be stable if the two tangents to the double point are imaginary,
and unstable if the two tangents are real.
We should note that the curve is symmetrical with respect to the two
axes of the coordinates and that the two double points, which are symmetri-
cal to each other with respect to the origin, do not correspond to two
periodic solutions which are actually different.
The double points may only be located on one of the axes of the coordin-
ates, so that they will be obtained by setting
If we set
1 k
C == -7^ -♦-■-+ ^I,
the curve F' = C passes through the origin and has a double point. The tan-
gents to the double point are given by the equation /358
Therefore, if
4 3.
, ;- ,u3 -t- uY cos 01 = o.
4 ^ „^
^7-- M^?>I^Y (1)
the tangents are imaginary. If
FY> ^-- -i-F?;- -J^T.. (2)
the tangents are real. Finally, if
the tangents are again imaginary.
The coefficient B is positive. I wrote the preceding inequalities
also assuming that y is positive. If y were negative, we would only have to
change o) into oi + tt.
The double point at the origin corresponds to the solution of the first
type, i.e., to planet A of M. Darwin. It may be seen that this solution is
stable when the inequalities (1) or (3) hold, and is unstable when the in-
equalities (2) hold.
Let us now study the double points which may be located on the line
u = 0.
353
If we set w = 0, the function F' becomes
F'= -,
i/>--^.;'
-tjta -1- iJt2:i(P + y)= C.
(4)
Keeping k constant, if we vary xi from to k, we find that the maxima and
minima of F' are given by the equation
(5)
(A--T,;»
■f(|3 + y) = o.
Which has a solution if the inequality (3) holds, and does not have a solu-
tion in the opposite case.
Therefore, if the inequality (3) does not hold, the function F' is con-
stantly decreasing if it holds. The function F' first increases, reaching
a maximum, and then decreases.
This maximum corresponds to a double~point located on the line u = 0./359
or rather to two double points which are symmetrical with respect to the
origin.
However we must determine how we may obtain these double points for a
given value of the constant C. Equation (5) provides us with xi as a func-
tion of k. We must deduce xi from it as a function of C.
However, equations (4) and (5) may be written
from which we have
dC _dV^_^dl^^^^dlL,
dFt " dXi "*" dk dxx dk dx,
dx\ dk dxi dx,
Neglecting terms containing y. we have
dF^ rfP
dk dx,
from which we have
dF'
dk
«"F'
d^r
dk dxi dx]
= o;
dx\ - {k-x,}' \8/
and
354
dx^ ' dxi
It results from this tliat xj is a constantly decreasing function of C.
For a value of C, we have only a maximum at the most, i.e., we have
at the most two double points which are symmetrical to each other with
respect to the origin on the line o) = 0.
Let Cq be the value of C which satisfies the double equality
„ 1 k
""o = ^ + -^- -t- I".
We shall see that, for C > Cg, there will not be a double point on the line
(0=0 and that, for C < Cq, there will be two of them. /360
The same discussion may be applied to the case of double points located
on the line co = ir. The values of xj will be given by the equation
(5')
which has a solution if the inequalities (2) or (3) hold.
If Ci is the value of C which satisfies the double equality
„ 1 k
X-j — ---■- H( ? — */) = o.
the condition for which there are two double points on the line
to = 7r,
is C < Ci.
We would like to point out that C^ > Cq , that Cq is the value of C
for which one passes from inequality (2) to inequality (3) , and that Ci is
the one for which we may pass from inequality (1) to inequality (2) .
When compiling the curves, we would readily find that the tangents are
real for the double points located on u = 0, and that they are imaginary
for the double points located on u) = tt.
355
\^
we may therefore sum up our results as follows:
First case
c > c,.
/361
The inequality (1) holds.
The solution of the first type (planet A) is stable.
There is no solution of the second type (orbit with double point).
Second case
The inequalities (2) hold.
The solution of the first type becomes unstable.
There is a solution of the second type which is stable.
Third case
Ce
the points where the four collisions occur.
We may specify these four points arbitrarily, provided that they are
located on the same line passing through F.
We must construct two ellipses E and Ei which intersect at Qi and Q^.two
360
ellipses E' and E'^ which intersect at Q2 and Q3, two others E" and EV
which intersect at Q3 and Qi^, and finally two others E'" and E'V which
intersect at Qi+and Qj .
The orbit of P is composed of arcs pertaining to the four ellipses
E, E' , E", S" , and the orbit of P^ is composed of arcs pertaining to the
four ellipses E^ , E\, E'{, E'|' .
We shall specify the energy and area constants arbitrarily. These /367
constants must be the same for the interval between the first two colli-
sions (orbits E and E^) for the following interval, and for all the other
intervals. According to the statements presented in the preceding section,
this is the only condition which must be fulfilled.
In order to compile E and E^ , we shall proceed as follows. Let us
consider the motion of three bodies. Since we assume p = 0, this motion is
Keplerian, and the central body may be regarded as being fixed at F. We
know the total energy of the system. The two planets P and Pj must leave
the point Qj simultaneously in order to arrive at the point Q2 simultane-
ously. When P and Pi go from Qi to Q2, the true longitude of P increases
by (2m + 1)tt, and that of P^ increases by (2mi + l)Tr. We may still specify
the two integer numbers m and mi arbitrarily. The problem has then been
completely determined. It should be pointed out that the inclination of
the orbits does not intervene. In order to resolve this, we may assume
planar motion. The problem can always be resolved. We need only apply the
principle of Maupertuis, and Maupertuis action, which is essentially positive,
always has a minimum.
We must now determine the planes of the two ellipses. We know the
area constants. We therefore know the invariable plane which passes through
the line FQxQ2- The areal velocity of the system is represented by a
vector perpendicular to the invariable plane, whose magnitude and direction
we know. It is the geometric sum of the areal velocities of the two
planets, represented by two vectors whose magnitude we know, since they
equal, respectively, mp and mip, where m and mi are the masses of the two
planets and p is the common parameter of the two ellipses E and Ei. We may
therefore compile the directions of these two base vectors which are perpen-
dicular to the plane of E and to the plane of Ei , respectively.
The terms E' and E^ , E" and E'j", ..., may be determined in the same
way.
389. Let us now assume that all of the successive collisions occur
at the same point Q. The period will be divided into as many intervals as
there will be collisions. Let us consider one of these intervals during
which the two planets describe the two ellipses E and Ei. As in the pre- /368
ceding section, we will specify the energy constant and the area constant
which must be the same for all the intervals. We must construct E and Ei.
361
K^
Let us assume that during the interval under consideration the planet
P has performed m complete revolutions, and that the planet Pj has completed
mi complete revolutions. We can arbitrarily specify the two whole numbers
m and mi- Since we know these two whole numbers, we know the ratio of the
major axes. Since we know, on the other hand, the energy constant, we also
know the major axes themselves.
On the other hand, we know the area constant. Consequently, we know
the vector which represents the areal velocity of the system. This vector
can be decomposed an infinite number of ways into two base vectors which
represent the areal velocities of P and Pj. We shall arbitrarily specify
this decomposition. If we know the two base vectors, we know the
planes of the two ellipses and their parameters. The orientation of each
of these ellipses in its plane remains to be determined. We will determine
it by passing the ellipse through the point Q.
Summarizing, we can arbitrarily specify:
1. The point Q and the number of intervals;
2. For all the intervals, the area constant and the energy constant;
3. For each interval, the whole numbers m and mi and the decomposition
of the areolar vector.
In order to make the problem tractable, these arbitrary numbers must
satisfy certain inequalities which I will not describe.
390. Let us disregard the exceptional case where all the collisions
take place along the same line or at the same point, and let us consider the
case of motion in a plane. Let Qi, Qz, • • • , be the points where the succes-
sive collisions take place. We will arbitrarily specify the energy constant
and the area constant which must be the same for all the intervals.
Let us consider one of the intervals, for example, the one where the
two planets pass from Qi to Qz- We will arbitrarily specify the magnitude
of the radius vectors FQi and FQ2, but not the angle between these two radius
vectors, nor the duration of the interval.
We know that in this interval the difference in longitude of the two /369
planets has increased by Zmir. Let us arbitrarily specify the whole number
m.
Since we know this whole number, the two lengths FQi and FQ2, as well
as the two energy constants and the area constants, we have everything
needed to determine the orbits E and Ej. This means that the principle of
Maupertuis must be applied. However, the Hamiltonian action must be defined
as was done in No. 339 and the Maupertuis action must be derived according
362
to the procedure of Nos. 336 and 337. Unfortunately, this Maupertuls
action is not always positive and therefore one Is not certain that it
always has a minimum.
Summarizing, we can arbitrarily specify:
1. The number of intervals and the lengths FQj^ , FQ2, ...;
2. The area constants and the energy constants;
3. For each interval, the whole number m.
The collision orbits obtained in this way are all planar . Among
the periodic orbits of the second kind which reduce to these collision
orbits for y = 0, there are certainly some which are planar. It is also
possible that there are some which are not planar for p > 0, and only
become so at the limit.
391. Let us now see how one may demonstrate the existence of periodic
solutions of the second kind which, in the limit, reduce to the collision
orbits which we constructed above.
Let us now consider one of the collision orbits and let tg be a time
before the first collision and t^ a time between the first and the second
collisions. In the same way, let t2 be a time between the second and the
third collisions. For the discussion I will assume that there are three
collisions. I will call T the period in such a way that at the time tg + T
the three bodies appear in the same configuration as was the case at the
time tg.
As the variables, I will take the major axes, the inclinations and the
eccentricities, and the differences of the mean longitudes, the longitudes
of the perihelia and the nodes. In all, there are eleven variables. The
orbit is regarded as periodic if the three bodies have the same relative
configuration at the end of the period.
Let Xj , Xg , ..., X]j be the values of these variables at the instant
tQ for the collision orbit under discussion and consequently for y = 0. /370
Let X. be the values of these variables at the time ti for this same colli-
sion orbit, x. their values at the time t^, and x^ their values at the time
to + T. One will have
■rf = x^ H- idht:
where m^ is a whole number which must be zero for the major axes, the eccen-
tricities and the inclinations.
363
\^
Let us now consider an orbit which is slightly different from the col-
lision orbit. Let us assign a very small value to y, but different^
from zero. In this new orbit, our variables will have the values x^ + S^
at the time to. x^ + B^ at the time t, . x? + B^ at the time t^ and finally
X? + b| at the time tQ + T + t.
The condition for which the solution is periodic with period T + t is
P? = ??•
Assuming y = 0, in order that a collision occurs between the time to
and the time ti, the variables bJ must satisfy two conditions.
Let
be these two conditions.
Let us set
it can be seen that the gO's are holomorphic functions of the yO's and of
y. By applying the principles of Chapter II, i^ can be shown that the same
holds for the B^'s.
In order that there be a collision between the times ti and t2 (assuming
that y = 0), two conditions are necessary, which I may write as follows
/■(?!)=/.(?;)= °- (1)
Replacing the b}'s in relationships (1) by their values as a function
of the Y°'s and of y, and then setting y = 0, I obtain
0.(y?)-''-'(t?)-o-
Let us then set
I find that the BJ^'s and the B^'s are holomorphic functions of the y^'s ^371
and of y. The same holds true for the yj's, and consequently for the bJ's.
Finally, in order that there be a collision between the times t2 and
to + Tt, two conditions are necessary which I may write as follows
364
If the 32 rg 3j.g replaced by their values as a function of the y^'s
i
and of y, and if we then set y = 0, they become
I may set
■')i(r,')=TiF. M{t\)-=ll\^, H=ll (^- = 3, .... II)
and I then find that the bJ's, the e^'s, and the sj's are holomorphic func-
tions of the Y?'s and of y. In the same way, the 6|'s are holomorphic func-
tions of the Y?'s, of y, and of t.
The relationships bJ = 3° are therefore equations whose two terms are
holomorphic with respect to the yj's, y, and t. These equations could be
discussed in the same manner as in Chapter III. The existence of solutions
of the second type could then be demonstrated.
I do not believe that this is necessary, because these solutions
deviate too much from the orbits traversed in actuality by celestial bodies.
365
\^
CHAPTER XXXIII
DOUBLY ASYMPTOTIC SOLUTIONS
Different Methods of Geometric Representation
392 In order to study doubly asymptotic solutions, we shall confine ^372
ourselves to a very special case, that of Section No. 9: Zero mass of the
perturbed planet; circular orbit of the perturbing planet; zero inclinations.
The three-body problem then has the well-known integral called the Jacob i
integral .
Returning to No. 299 devoted to this problem from Eo. 9, we must dis-
tinguish between several cases. We saw on page 159 that we must have the
following inequality
■?«
r, r, 2
(1)
2
We then distinguished between the case in which m^ is much smaller than m^, and
in which -h is sufficiently large (page 160) . We saw that the following
curve
V .--(?'+o«)=-A (2)
2
may be broken down into three closed branches which we have called Ci , Cg
and C3. Therefore, in view of the inequality (1), the point 5, n must always
remain inside of Ci , or always inside of C2, or always outside of C3 (?, n
are the rectangular coordinates of the perturbed planet with respect to the
moving axes) .
We shall assume below that the value of the constant -h is large enough
for curve (2) to be broken down into three closed branches, and that the
point 5, n always remains inside of C2. In this way, the distance T2 ^''°'^
the perturbed planet to the central body may vanish, but this is not true 2J11
for the distance ri between the two planets.
This hypothesis corresponds to the following hypothesis, which we formu-
lated on pages 199 and 200 — i.e., the curve F = C has the form shown in
Figure 9, and the point xi, X2 remains on the utilizable arc AB.
We shall employ the notation given in No. 313, and we shall introduce
the Keplerian variables L, G, 1, g. However, these Keplerian variables
may be defined in two ways. Just as in No. 9, we could relate the perturbed
body to the center of gravity of the perturbing body and of the central body.
366
and we could consider the oscillating ellipse described around this center
of gravity. However, it is preferable to refer the perturbed body to the
central body itself, and to consider the oscillating ellipse described
around this central body.
These two procedures are equally legitimate. We saw in No. 11 that
the body B may be related to the body A, and the body C may be related to
the center of gravity of A and of B. It is apparent that we could also
refer C to A, and B to the center of gravity of A and C. If A represents
the central body, B the perturbing body, and C the perturbed body, it can
be seen that the first solution is that which was adopted in No. 9. It
may also be seen that in the second solution, which we shall adopt from
this point on, the two bodies B and C are both related to the central body
since ~ due to the fact that the mass of C is zero — the center of gravity
of A and C is at A. o j
We then have
^^ '1 2 / 1 — JJl
where y and 1 - p designate the masses of the perturbing body and of the
central body, ri designates the distance between the two planets, 1 desig-
nates the constant distance from the perturbing body to the central body,
and r2 designates the distance of the perturbed body to the central body.'
Just as in No. 313, we shall set
r, ^-. L — G, a-, = L + G,
■iy-i^- l — S + i, ■iri=l-i-ff~r,
^ ' ' i /l — - fZ
I would like to stress the following important point. It can be seen that /374
the function Fi always remains finite in the region from which the point 5 ,
n cannot leave.
We shall employ the method of representation given on page 200 , and we
shall represent the configuration of the system by the point in space whose
rectangular coordinates are
7_ _ , I /. r, o.
When xi, X2, yi, y2 vary in conformance with differential equations,
the point X.'y, Z will describe a certain trajectory. When y2, which in-
creases constantly, reaches the value 2it, the point X, Y, Z — which has
moved to Ml ~ will again be located on the half-plane Y - 0, X > 0.
The point Mi is then the consequent of M, according to the definition
given in No. 305. Since y2 is always increasing, every point on the half-
plane has a consequent and an antecedent. There is only an exception for
368
\^
very small X2 — i.e., for points on the half -plane which are very far
from the origin, or very close to the Z axis.
We shall have an integral invariant, in terms of the meaning attribu-
ted to this word in No. 305. Let us try to formulate this invariant.
Due to the fact that the equations are canonical equations , they have
the following integral invariant
I ilTidXidyicIy^.
X2
Let us set z = — , and let us select F , z, y^, y2 as new variables.
^1
The invariant will become
'r; f/PV/if/yirfi-j _ (' x]dF'dzdy^dYt
■ ''' _ I ^1 "•' dzdy^d
dV tfi' I ^1 /^i -t- x"5 /(.
rf.r, dxi
We may deduce the following triple invariant from this quadruple in-
variant (due to the existence of the integral F' = C)
x\dzdridYi
rx\dzd y,dy^
In this triple integral, we assume that x,, ^2, «, = — ^— , «, = _ 5— are
axi dx^
replaced as functions of z, yj^, y2 by means of the equations
j?j = .r, ;, F'=C.
Let us now take the variables X, Y, Z, and let us employ A to desig- /376
nate the Jacobian of X, Y, Z, with respect to z, yi , y2. The invariant will
become
r x\d\d \dl
J (ar|/ii-+-^,/!,)i'
Let us set
R=^ ^ .^' — , Z =
from which it follows that
X =. R cos^», Y =^ R sin^,.
Let US again set D ^-[(R -,)• + z*j[(R + ,)^4-Z»]-
369
\^
A simple calculation provides the following
8 /.-(; + 4)
Our invariant may therefore be written
r 8t> /^'(iT T) d\ d\ (17.
The principles presented in No. 305 enable us to deduce the following
invariant, in the sense of No. 305
8,r7v'^(--H4:
r^iihS
.r,/ii-+- T, n,
d\ d'L.
n2 and R play the role which fi and p played in the analysis of No. 305.
The tern under the sign / is essentially positive, except for very
small X2 — i.e., for points of the half -plane which are very far from the
origin, or very close to the Z axis.
393. This fact (that a point will no longer have a consequent if it
is too far, or if it is too close, to the Z axis) could cause some difficulty,
and it would be advantageous to avoid this difficulty by whatever method.
We could employ the statements presented in No. 311, and we could re-
place our half-plane by a simply connected curve on a surface. We shall
choose this curve on a surface in the following way.
If X2 is very small, the eccentricity is very small, and the two planets
turn in the opposite direction. The principles presented in No. 40 are
applicable, and we may affirm the existence of a periodic solution of the /377
first type which will clearly satisfy the following conditions: The quanti-
ties
/z^cosj'2, /x^sin/,, xu cos/,, sinj,
are periodic functions of the time t. These functions depend on y and on
the energy constant C. They may be developed in powers of y; the period
T also depends on y and on C. The angle yi increases by Zir when t increases
by a period. Finally*, V^ cos ya and yx^ sin y2 are divisible by y, so
that we have X2 = for y = 0.
With our method of representation, this periodic solution, which I
have called a, is represented by a closed curve K. Since X2 is very small
when y is very small, this curve is displaced very little from the Z axis.
370
It may be stated that it is displaced from it very little, in the same way
that a circle having a very large radius is displaced very little from a
straight line. Every point on the K curve is either very far from the origin
or very close to the Z axis.
Under this assumption, our curve on a surface S would have the curve
K for the perimeter, and it would be displaced very little from the half-
plane Y = 0, X > 0, except in the immediate vicinity of the curve K. It
would be very easy to conclude this determination in such a way that every
point on this surface would have a consequent on this surface itself. For
this purpose, if I designate an arbitrary trajectory by (T) — i.e., one
of the curves defined in our method of representation by differential equa-
tions — it would be sufficient that the surface S was not tangent at any
point to any of the trajectories (T) .
However, there is still another method, which does not basically differ
from the first method. If we reflect on this a little, we will find that
this difficulty is similar to that in Chapter XII. We must therefore perform
the change in variables similar to that performed in No. 145.
Let us first set
and we then have
S ---^•'■.i-h.r;^-, -hpS,
where Si is a function of C'2. 12* x'l , yi. Let us then set /378
and finally
I should first point out that the canonical form of the equations will
not be changed when I pass from the variables xi , yi , X2, yz, to xi , yi, C2»
TI2, then to x'l, y'l , ^\^ t\\, and finally to x\ , y'^ , x'^, y'g.
I must now choose the function Si.
I know that F' is a holomorphic function of v/^-^'i ♦'os^j^i, v'ax, sfnjKi,
/axacosj^'j, y/axjsinjK- in the region under consideration. I would like it to
371
K^
remain a holomorphic function of the new variables
v'T^cos/',, /^sin/;.
For this purpose, I would like the old variables v/i^ gin yi ^° ^^ holomor-
i COS
phlc functions of the new variables V2x'^ 3^^ ^i ^^^ °^ ^'
To do this, we need only assume that Si is a holomorphic function of
and is divisible by x'^.
For our periodic solution a, I would like to have
tj = T/j — o, x', — xj = const.
Therefore, let
be the equations of the periodic solution. A, B, C are functions of yi
which are periodic of the period 2tt and may be developed in powers of y .
Then
C - -^^ will also be a periodic function of yi- Let xj be its
dyi
mean value. We may obtain another periodic function a such that
dy^. ilyx
We shall no longer assume that, for x'l = xj, the function ySi may be re- /379.
duced to
«-B$;-+-Ar„. (2)
This will be sufficient for the equations of the periodic solution
to be reduced with the new variables to
It is clearly possible to obtain a function ySi which may be developed
in powers of V^i ^l^ Yl. which may be divisible by x'l, and which at the
same time may be reduced to expression (2) for x'^ = x^
n
Let VIS adopt the new variables x*i,y'i, x'2, y'z'
372
v_/
The function F', which was holomorphic with respect to l/2xi *^°^ vi
' ^ sin ' '■*
V^x^j^^ 72* will also be holomorphic with respect to l/ix^ ^°^ y'l,
I COS
V2x'2g^^ 7 2 . In addition, since one of the solutions of the differential
equations is
we must have the following relationships for 5*2= n'2 = 0, x'l = xP
^F' _ d¥' _ rfF' _
t^^, - c/v; " ^ - "■ (3)
For small values of C'2 and n'2 » F' may be developed in powers of 5*2
and n'2. In view of relationships (3), for x'l = xj , the terms of the first
degree in this expansion will vanish, and the terms of zero degree will be
reduced to a constant which is independent of y^.
This constant can be nothing else than the energy constant C, so that
the conditions 5'2 = n'2 = 0, x'l = xj may be replaced by the following con-
ditions
?i = 7)', = 0, F'=C.
Thus, for F' = C, the terms of the first degree in ^2 and r/2 will vanish
in the expansion of F',
The difficulty arises from the fact that F' and Fi include terms of
the first degree in /380
and that, consequently, the derivative -~ includes terms — = — which be-
come infinite for X2 = 0.
This difficulty no longer exists now. We no longer have terms of the
first degree in C'2, n'2- Therefore the derivative ^^ remains finite,
dF' dFn
even for X2 = 0,and — -, which differs very little from -ri; always retains
CIX2 CIX2
the same sign. Therefore, with our new variables which only differ from
the old variables by very small quantities on the order of m, we shall con-
stantly have
dx\
-dr>°-
373
\^
With our new variables, let us formulate a convention which is sindlar
to th!t given in the preceding section, and let us -present the confxgura-
tion of the system by the point in space whose coordinates are
v/7; COS7J ^ Y ^ ___^/{:!.^'!lj:L_ ,
/r', -<-\iJ^'i-2 A; cosy,
dx'2
Everything which we have stated still holds . However, since -^
can never vanish. PVPrv noint on the half -plane, without exception, will
have a consequent .
It may now be stated that the integral invariant is ^^^^y^ .P°^^^^^^- ,
There ca^ only be some question of doubt for the denominator which, wxth the
same variables, was xjni + xzng and which now would be
Which - assuming that F' is a function of the following four variables
/381
may be written
I A, dF' , dr „ d?' , dr\
-iK'^-du^'-d^r^'Wr^'-^WJ-
In this form, it may be readily seen that the denominator ^^ ^°\f^°J^^ll
with respect to the 5' 's, the n"s, and y. However, for y = 0, F may be
reduced to
(x\ -i-a?;)»
and it may readily be shown that the denominator is always positive. It
will still be positive for small values of y.
394. In the following statements, we shall adopt the variables defined
in the preceding section. We shall remove the accents which have become
useless, and we shall write F, Xi and y^ in place of F' , x . and yi- We
then have the integral invariant (in the sense of No. 305)
374
"J D ■ —dV 7/F '^^ '^2
from which we have ' ''•'"s
£>-[(.X-r)M-Z*J[(X-t-,}M-Z'].
I would first like to note that this integral invariant, which is
always positive, remains finite when it is extended over the entire half-
plane.
If l/(X - 1)2 + z2 is an infinitely small quantity of the first order,
the numerator x2 Vz(z + 4) is an infinitely small quantit y of the second
order, and the same holds true for D. If l/(X - 1)2 + z2 is an infinitely
large quantity of the first order, the numerator remains finite, while D is
very large of the fourth order. All of the other quantities remain finite.
I shall call Jg the value of the invariant J extended over the entire
half-plane.
The periodic solutions and the trajectory curves which represent them
are characterized by the fact that these curves intersect the half-plane at
points whose successive consequents are finite in number. For example, let
us refer to No. 312 and, in particular, to Figure 7 shown in page 195.
In this figure, the closed trajectory which represents a periodic /382
solution intersects the half-plane at five points Mq, Mi. Mg, M3 , M^, each^
of which IS the consequent of the others. For purposes of brevity I
shall call such a system a system of periodic points or a periodic 'system .
Two systems of as^ptotic solutions correspond to each unstable, periodic
r J^o^ ^"T^t solutions are represented by trajectories (in the sense of
No. J12), and the total group of these trajectories forms what I have desig-
nated as asymptotic surfaces. The intersection of an asymptotic surface
with the half-plane will be called an asymptotic curve . Just as we saw in
Figure 7, page 195, four branches of asymptotic curves (MA, MB MP MQ) ~
each two of which are located in the extension of the other ~ lead to each
of the points M^ of an unstable periodic system.
There is an infinite number of asymptotic curves, because there is an
infinite number of unstable, periodic solutions and, consequently, an
infinite number of systems of unstable periodic points, even if we confine
ourselves to solutions of the first type which we defined in Nos. 42 and 44.
A distinction may be drawn between asymptotic curves of the first
375
K^
family and of the second family, depending on whether ^^^ --"P^f^^^
characteristic exponent is positive or negative Curves "J^^^^ ^^^^^^^
family are characterized by the following P-P"^/' .^f ,f; Jf^ery
of an arbitrary point is very close to a periodic P^^^^ xf^ eonleouent,
laree For curves of the second family, it would be the n— consequent
and n;t the rh antecedent, which would be very close to a periodic point.
On the figure shown on page 195 . the curves MA and MP belong to the
first fallly, Ld the curves MB and MQ belong to the second family.
These asymptotic curves may be regarded as invariant curves in the
Iti-nT'^u^:^ l^n^S^^^ ^^^^ --fit
M3A3, M4A4, this total Sroup wii ' ^f 5 ^^d if we designate the 5p—
we only consider the consequents m groups o^ ^' '^ ^ ,, ^hp nth conseauent,
conseauent which it has been called up to the present, as the p— consequent ,
iriraprarent that only the curve MqAoA^ under consideration will be an in-^
variant curve.
T wo curves of the same family can not intersect. These two curves will
Pnd at the same periodic point - for example, the point Mo- These two
curves will coincide (since MqAq with its extension MoPq is the only curve
of the first family which passes through Mo), and we must determine whether an
asymptotic curTe can have a' double point. The question has been answered
in the negative (No . 309 , page 186) .
Or these two curves will lead to two periodic points of the same
periodic systL - for example, to the two points Mo ^^^ ^^ -^1',% 'ZT'
which would then be MqAq and M.A^ , had a point m common Q the ^P-^nte
cedent of Q would have to be very close to Mq for very ^^^S^P ' J^J^^"^^^^
Tould belong to MqAq, and it would have to be very close to M^ at the same
time because Q would belong to MiAi . This is absurd.
Or finally the two curves would lead to two points belonging to two
different periodic systems. For example, let us assume that the two curves
belong to ?he first family, and that Q is their point of intersection.
For very large n. the n^l^ antecendent of Q would have to be very close
to one of the points ;f the first periodic system and one of the points of
the second system at the same time. This is also impossible.
Conversely, ^h.rp i. no reason tha t two as ym ptotic curves of different
families cannot intersect .
Let S and S' be two unstable periodic solutions, let T and T' be the
376
corresponding closed trajectories, and let P and P' be the corresponding
periodic systems.
Let H and Z ' be two asymptotic surfaces wfiich pass through T and T' ,
respectively, and which intersect the half-plane along two asymptotic
curves C and C' — one belonging to the first family, and the other belonging
to the second family.
What will happen if C and C' have a point in common Q? The two sur-
faces E andE' will intersect along a trajectory x, which will correspond
to a special solution 0. The trajectory x will belong to two asjmipototic
surfaces, so that for t = -« it will closely approach T, and for t = +«> it
will closely approach T' . For very large n, the n£ll antecedent of Q will be
very close to one of the points of system P and its n£k consequent will be /384
very close to one of the points of system P'.
The solution a is therefore doubly asymptotic .
There is nothing absurd in any of these results.
We must distinguish between two cases, however. The two solutions S
and S' coincide, so that x first closely approaches T = T' , then
recedes farther away from it, and again closely approaches this same tra-
jectory T = T'. I could then state that the solution a is homoclinous . Or,
S differs from S' , and T differs from T' ; I may then state that a is hetero-
clinous .
The existence of homoclinous solutions will be demonstrated very
shortly. The existence of heteroclinous solutions remains doubtful, at
least in the case of the three-body problem.
Homoclinous Solutions
395.
. At the end of No. 312, we found that "the arcs AqAs and B0B5
intersect". However, the arc A0A5 belongs to the curve MqAqAs which is an
asymptotic curve of the first family, and the arc BQB5 is part of the curve
M3B0 which belongs to the second family.
The line of reasoning is general, and we must conclude that the two
asymptotic surfaces which pass through the same closed trajectory must
always intersect beyond this trajectory. The asymptotic curves of the first
family which lead to the points of a periodic system always Intersect the
curves of the second family, which lead to these same points.
In other words, on each as3miptotlc surface there is at least one doubly
asymptotic, homoclinous solution. We shall see very shortly that there is
an infinite number of them, but we shall now show that there are at least two
377
of them.
For this purpose, let us turn to the figure shown on page 195. Follow-
ing the line of reasoning in Nos. 308 and 312, we find that tLe integral
invariant J extended over the quadrilateral AqBqAsBs must be zero. It is
for this reason that this curvilinear quadrilateral cannot be convex, and
that the opposite sides A0A5 and B0B5 must intersect. Let Q be one of the /J«l
intersection points of these two arcs. We should note that the point Bq
was chosen arbitrarily on the asymptotic curve MAq . If we place the point
An at the point Q itself, this point Aq will also be located on the curve _
M3B0 and will coincide with the point Bq . If the two points Aq and Bq coin-
cide, the same will hold true for their five consequents A5 and B5.
The quadrilateral A0B0A5B5 will therefore be reduced to the firgure
formed by two arcs of a curve having the same end points. This figure can
not be convex, since the integral invariant extended over the quadrilateral
must be zero. Therefore the two arcs AqAj and B0B5 must have points in com-
mon, other than their end points.
There will therefore be at least two different intersection points
(a point and an arbitrary consequent of it are not regarded as being differ-
ent) .
There will therefore always be at least two doubly asymptotic solutions.
Let us assume that the points Aq and Bq coincide, and let us extend
the arcs AqAj and B0B5 up to the first point at which they touch Cq. We
will have thus determined an area which will be convex this time (from the
point of view of Analysis situs) and which will be bounded by two arcs
which are apart of the two arcs AqAj and B0B5, respectively, having the
same end points — i.e., Aq = Bq and Cq.
Let ao be this area, and let ct^ be its v^ consequent. The area a^ —
like an ~ will obviously be convex and bounded by two arcs of a curve --
one belonging to the first family, and the other belonging to the second
family.
The integral J will have the same value for Oq and an- L et j be this
value. Since the value Jq of the integral invariant for the entire half-
plane is finite, following the line of reasoning presented in No. 291, we
will find that, if
^ Jo
J
the area ao will have a part in common, at least with p of the areas
M, ■'-1, ■ ■ •> 'n .
378
V-/
Since n cannot be taken arbitrarily large, I may stipulate the following
result:
Among the areas ot^, there is an infinite number o f them which have a
part in common with ctp^ ' —
How may it happen that qq has a part in common with 0^? /386
The area ao cannot be entirely within 0^, since the integral invariant
has the same value for the two areas. For the same reason, the area an
cannot be entirely within Oq. Neither can the two areas coincide. If one
part of an asymptotic curve (for example, belonging to the first family)
coincided with its n^ consequent, the same would hold true for its p£B.
antecedent, no matter how large p may be. However, if p is large, this plk
antecedent is very close to the periodic points, and the principles formu-
lated in Chapter VII will demonstrate that this coincidence does not occur.
We must therefore assume that the perimeter of uq intersects that of
an. However, the perimeter of a^ is composed of an arc AqHoCq belonging to
the curve MqAqAs of the first family, and of an arc
belonging to the curve M3B5B0 of the second family.
In the same way, the perimeter of a^ will be composed of the arc AnMnCn,
the nta. consequent of AoHqCo, which will belong to the same asymptotic curve
as AqHoCo — i.e., to a curve of the same family — and it will also be com-
posed of the arc AnKnCn, the nth consequent of AoKqCo, which will belong to
the same asymptotic curve as AqKoCq ~ i.e., to a curve of the second family.
Due to the fact that two curves of the same family cannot intersect, it
is necessary that AqHoCo intersect AnKnCn, or that AoKqCo intersects AnHnCn.
However, if the two arcs AoKqCq and AnHnCn intersect, their nSll antecendents
A_nK_nC_n and AqHoCq will equally intersect. It is therefore necessary that
AqHoCo intersect the nth consequent, or the n£]l antecedent, of AoKqCq.
However, the arc AoKqCq, all of its antecedents, and all of its conse-
quents will belong to the same invariant curve of the second family, which
was shown in the figure on page 195 by the total group of curves M3B0, M1B3,
Mi^Bi , M2B4, M0B2.
Ihe arc AoHqCq is therefore intersected an infinite n umber of times by
this group of curves . " ' —
The two surfaces Z and E' which passed through the closed trajectory T
therefore have an infinite number of other intersection curves.
379
-n.prpfore. on the surface Y. there T^ an Infinite number of double
asymptotic, homoclinous s olutions.
• e.d.
396. Let AoHoCo be an arbitrary arc of our asymptotic curve of the
first family, and Lt us assume that this arc intersects an asymptotic
cirve ofthe second family at two end points A, and Co- It may be stated
thT there will always be other points of intersection with the curve of
the second family between these two points Aq and Cq.
Let AqKoCo be the arc of the curve of the second family which unites
these two points Aq and Cq-
Either the two arcs AqHoCo and AoKqCq have points in common other than
their end points, in which case the theorem has been proven.
or these two arcs do not have a point in common other than their end
joints An and Cn. The two arcs then bound an area ag which is similar to
'that whtch Se considered at the end of the preceding section Ihe same line
of reasoning may then be applied, and we may conclude that the arc AoHqCq
intersects the curve of the second family an infinite number of times.
Therefore, there is an infinite number of other points on an asymptotic
curve of the first family, between two arbitrary points of intersection with
the curve of the second family.
nn pn arbitrary asymptotic surface, between two doubly asympto.tl^_ar^
bitrary solutions, there is an inf inity of other solutions.
We may not yet conclude that the doubly asymptotic solutions are ever^-
where dense on the asymptotic surface, but this seems very likely.
The points of intersection of two asymptotic curves may be divided into
two categories. The asymptotic curve may be traversed in two ^PP^^^;^^/"^^:
tions. We assume that this direction is positive, if we proceed from a point
tolls consequent. L et A be a point of intersection of the two curves, and
let BAB', CAC be two asymptotic curve arcs intersecting at A. Let us assume
that BAB' belongs to the first family, and CAC belongs to the second family,
and that -- when following the curves in the positive direction - one V^°-
ceeds from A to B' . and from A to C . Depending upon whether the direc- /388
tlon AB- is to the right or the left of AC', the intersection point A will
belong to the first or to the second category.
Under this assumption, let AqHoCq be an arc of the first family, inter-
sected at Ao and Co by an arc AqKoCq of the second family. No -^"er what
category Aq and Cq belong to, the group of two arcs AoHoCoKqAo will form a
closfd curve. If the two arcs have no other point in common except their
end points, this closed curve does not have a double point and defines an
380
area Og. If the two arcs had points in common other than their end
points, and if, for example, the two arcs AoHqDqH^Co, AqKoDoK^Cq inter-
sect at Dq, we may replace the points Aq and Cq by the points Aq and Dq
located between Ag and Cq , and the arcs AqHqCo , AqKoCq by the two arcs
AoHqDo and AqKoDq. This may be continued until we arrive at two arcs
which have no point in common other than their end points.
Let us assume that the two arcs define an area clq. According to the
statements we have just presented, the arc AqHoCo must intersect the asymp-
totic curve of the second family an infinite number of times. Therefore,
the curve of the second family must penetrate within qq an infinite number
of times, and it must leave it an infinite number of times. It may pene-
trate it or leave it only by intersecting AqHoCo, because it cannot inter-
sect AoKqCq which also forms a part of the curve of the second family. It
is apparent that points through which it will penetrate into the area, and
the points through which it will leave the area, will not belong to the same
category.
Therefore, between two arbitrary intersection points of two curves.
there i s an infinity of other points belonging to the first category, and
an infinity of other points belonging to the second category .
Let us employ (1), (2), (3), ..., to designate the successive points at
which the curve of the second family and the arc AqHoCo meet, taken in the
order in which they are encountered proceeding along the curve of the second
family in the positive direction. They will belong to two categories in
succession. Let us study the order in which they are encountered proceeding
along the arc AqHoCo.
This order cannot be completely arbitrary, and certain successions are
excluded — for example, the following: /289
(7.ni), {xp)^ (rL?n H- I), (?./>-:- I)
(iin+ I) (2/,), {■im), {,-).p .- \)
('•'«)> (2/J I- I}, {im+ i), (,,;,)
(i"0. {r>.p), (2m -I), (2/?- I)
as well as the same inverse successions, and the similar successions where
2m + 1 and 2p + 1 are replaced by 2m - 1 and 2 p - 1.
397. When we try to represent the figure formed by these two curves
and their intersections in a finite number, each of which corresponds to a
doubly asymptotic solution, these intersections form a type of trellis, tissue,
or grid with infinitely serrated mesh. Neither of the two curves must ever cut
across itself again, but it must bend back upon Itself in a very complex
manner in order to cut across all of the meshes in the grid an infinite
number of times.
381
\^
The complexity of this figure will be striking and I shall ^ot ^^en
try to draw it. Nothing is more suitable for providing us with an idea of
the complex nature of the three-body problem, and of all the Problems of
dynamics in general, where there is no uniform integral and where the Bohlin
series are divergent.
Different hypotheses are possible.
1. We may assume that the group of points of two asymptotic _ curves Eq,
or the group of points in the vicinity of which there is an infinite number
of points belonging to Eq - i.e., the group E'o, the derivative of Eq"
occupies the entire half -plane . We would then have to conclude that insta-
bility of the solar system exists.
2. We may assume that the group E'o bas a finite area and occupies a
finite region of the half-plane, but does not occupy it completely. Either
one part of this half-plane remains outside of the meshes of our grid, or
a "gap" remains within one of these meshes. For example, let Uq be one of _
these meshes bounded by two or more asymptotic curve arcs of the ^wo families
Let us compile its successive consequents, and let us apply the procedure /390
presented in No. 291. Just as on page 145, let us formulate the following
u„ u'„ u^, u;, u;, ..., k.
If it is finite, the area E will represent one of the gaps which we
just mentioned. It would appear that we may apply the line of reasoning
employed in No. 294, and may conclude that this area must '^f ^J^'^^^^J^ °"^
of its consequents. However, this group E could be composed of a region of
finite area and of a group located outside of this region whose total area^
would be zero. According to page 151, we may only conclude that E^ (the Xii^
consequent of E) includes E, and that the group Ex - E has — ^^^^J^^^^
same way, the groups E - E_x, E.^ - E_2X. •••. ^-n\ " E_(n+l)X will have area
z^o Z area of a group, we meai the value of the integral J extended over
this group), on thf other hand E_(,+0X is a part of E_,, . When n increases
indefinitely. E_„x tends toward a group e including every point which is part
of all the g;oups E_„, at the same time. The area of this group e is finite
and equals that of E. Finally, e coincides with its A-^ consequent.
3. Finally, we may assume that the group E'o has area zero.
It would then be similar to those "perfect groups which are not con-
densed in any interval".
398 We may represent the different intersection points of the two
curves in the following way. Let x be a variable which varies from -» to +",
when the asymptotic curve of the first family MoAq is followed, from the
point Mo up to infinity, and which increases by unity when we pass from one
point to its fifth consequent ~ from Aq to A5, for example (to clarify this
point, we shall assume that we are dealing with the conditions of the figure
382
shown on page 195). Let y be another variable which varies from +- to -»
when the curve of the second family K^B^ is followed from the point M3 up
to infinity, and which increases by unity when we pass from a point to its
fifth consequent.
The different intersection points of the two curves are characterized
by two values of x and y, and each of them may be represented by the point
on a plane whose rectangular coordinates are x and y.
We shall thus have an infinite number of representative points of the /391
doubly asymptotic solutions in the plane. An infinite number of other pointl"
may be deduced from each of these points. If the point x, y corresponds to
an intersection of the two curves, the same will hold true for the points
^"i-i,^-hi; .r + 9., ;^' + 2; ...: .r-t-n, y -i- n,
where n is a positive or negative whole number. In order to determine all
the representative points, it is sufficient to know all those which are in-
cluded in the region < x < 1, or in the region < y < 1.
We would also like to note that the order in which the projections of
these representative points will occur on the x axis will have no relation-
ship with the order in which their projections will occur on the y axis
■nils results in the following.
Let us consider several doubly asymptotic solutions. For t which Is
negative and very large, they will all be very close to the periodic solu-
tion, and they will appear in a certain order ~ some of them will be closer
to, and others will be farther from, the periodic solution.
All of them will then recede appreciably from the periodic solution, and
-- for t which is positive and very large — they will all again be very
close to It. However, th ey will then appear in an entirely different order ,
out of two solutions, if the first is closer than the second to the periodic
solution for t = —, it may happen that for t = +- the first is farther away
than the second from the periodic solution, but the opposite could also occur.
We have pointed this out in order to illustrate the great complexity of
the three-body problem, and to show how many different transcendents out of
all those which we know must be considered in order to solve it.
Heterocllnous Solutions
399. Do heterocllnous solutions exist?
As far as we can determine, if there is one of them, there is an infinite
number of them.
383
\^
Let Mo be a point belonging to a periodic system. Let MqAq and MqBo /392
be two asymptotic curves bordering upon this point Mq --one belonging to
the first family, and the other belonging to the second family. We have
just seen h^ tLse curves intersect, so that the doubly asymptotic, homo-
clinous solutions may be determined.
Now let Mb be a point belonging to another periodic solution. Let
M'oA'o, M'uB'o be two asymptotic curves, M'A'q belongs to the first family,
and M'B'o belongs to the second family.
Let us assume that M'qA'o intersects MqBo at Qq. This intersection
will correspond to a doubly asymptotic, heteroclinous solution.
However, if these two curves intersect at Qo , they will also intersect
at an infinite number of points Q^, the consequents of Qo-
I shall state this precisely. For example, I shall assume that the
periodic system of which Mq is a part is composed of five points Mq , Mi M2,
M, Mu Then the fifth consequent of an arbitrary point of the curve MqEo
will still be located on this curve, and in general — if Qo is on this
curve ~ the same will hold true for its ntk consequent Q^, provided that n
is a multiple of five.
In the same way, let us assume that the periodic system of "hich M'q
is a part is composed of seven points. Then, if Qo is on the curve M oAq,
the same will hold true for its nth consequent Q^, provided that n is a
multiple of 7.
Therefore, if the two curves have an intersection at Q , they will still
have an intersection at Q^, provided that n is a multiple of 35.
Let QoHoQn be an arc of MoBq, and let QoKoQn be an arc of M'q A'o . Due
to the fact that these two arcs have the same end points, together they will
form a closed curve. We may pursue the same line of reasoning as in No. 396
for this closed curve. We shall find that, if the two arcs have no other
point in common except their end points, this closed curve does not have a
double point, and defines an area which is similar to the area ag given in
Nos 395 and 396. If the two arcs have points in common other than their
end points, we may obtain two other arcs which are part of the two arcs
QoHoQn. QoKoQn which have only their end points in common and which define
an area similar to ag.
The same line of reasoning as was employed in Nos. 395 and 396 may be
used for this area ao, and we will find that an infinite number of other
points may be obtained on each of the two curves, between two arbitrary I39J
points of intersection with the other curve.
This line of reasoning shows that if there is one heteroclinous
384
■ solution, there is an infinite number of them.
400. If there is a heteroclinous solution, the grid of which we
spoke in No. 397 must be still more complicated. Instead of a single
curve MqAq bending back upon itself without ever cutting across itself,
and intersecting the other curve MqBq an infinite number of times, we
shall have two curves MqAq , M'gA'g which must intersect MqEq an infinite
number of times without ever cutting across each other.
In No. 397, we defined the group E'g with respect to the point Mq and
to the asjrmptotic curves MqAq, MqBo. We may also define a similar group
with respect to the point M'g and to two asymptotic curves M'qA'o, M'q^O'
If there is no heteroclinous solution, these two groups must be out-
side of each other; therefore, they cannot occupy the half-plane. If, on
the contrary, there is a heteroclinous solution, these two groups will
coincide. It may be seen that the existence of such a solution — if it
could be established — would provide an argument against stability.
In Chapter XIII we studied the series of Newcomb and Lindstedt, and
we showed in No. 149 that these series cannot converge for every value of
the constants which they contain. However, one question remains in doubt.
Could these series converge for certain values of these constants and, for
example, could it happen that the convergence occurs when the ratio 111. is
n2
the square root of a commensurable number which is not a perfect square
(see Volume II, page 104, in fine).
However, if a heteroclinous solution does exist, the answer to this ques-
tion must be in the negative. Let us assume that for certain values of the
ratio — the series of Newcomb and Lindstedt converge, and let us return to
our method of representation. The solutions of the differential equations
ni
which would correspond to this value of — could be represented by certain
trajectory curves. The group of these curves would form a surface, having
the same connections as the torus, and this surface would intersect our /394
half-plane proceeding along a certain closed curve C.
The group E'g which we just mentioned would have to be completely out-
side of this curve, or completely inside of it.
Let Mq and M'o be two points belonging to two different systems. If Mq
is within the curve C and M'g is outside of this curve, the group E'q with
respect to Mq would have to be entirely within it, while the group E'q with
respect to M'q would have to be entirely outside of it.
These two groups could not have any point in common, and no doubly
385
asymptotic, heteroclinous solution could exist, proceeding from Mq to M'q .
If we admit the hypothesis advanced in Volume II, page 104, which X
have just presented — i.e., if the convergence occurs for an infinite
number of values of the ratio ^, for example, for those whose square is
n2
commensurable ~ there would be an infinite number of curves C which would
separate the points belonging to different periodic systems. This hypothesis
is incompatible with the existence of heteroclinous solutions Cat least if
the two points Mq and M'q which we are considering, or the corresponding
periodic solutions, correspond to two different values of the number "_]_.)
Comparison with No. 225
401. Before trying to present examples of heteroclinous solutions, we
shall return to the example of No. 225, where the existence of doubly asymp-
totic, homoclinous solutions may be illustrated.
We set
. F =/)-(-9'— ■Jiiisin*- — (iE^'e'']
/396
In order to obtain the doubly asymptotic solutions, we must determine
the Intersection of these two asymptotic surfaces. It will be sufficient
for us to equate the two values of p and the two values of q.
L et us set
J„ ' + /»
U :r^1 log/.
We shall obtain
^ real part [J/e-''«+'.f] = o,
^- real part [Ue~a"*-i^]^-o,
387
K^
or, setting J = pe^",
where K is a whole number.
This is the equation of doubly asymptotic solutions.
In reality, this equation provides us with two different solutions,
one corresponding to even values of K, and the other corresponding to odd
values of K.
402. We may be surprised at not obtaining more than two doubly asymp-
totic solutions, when we know that there is an infinite number of them.
The following approximations should provide us with no more than a
finite number of doubly asymptotic solutions. How may this paradox be
explained?
In the preceding sections we saw that the different doubly asymptotic
solutions correspond in an infinite number to different intersections of a
certain arc AoHqCo with the different consequents of another arc AqKoCo.
Let us assume that the first of its consequents which encounters AoHqCo
is the consequent of order N. The number N will clearly depend on the con- ^397
stant e, and the smaller the constant is, the larger it will be. It will
become infinite when e is zero .
If we develop in powers of e and stop at an arbitrary term in the ex-
pansion, it is as though we regarded e as being infinitely small.
The arc AqHoCq no longer encounters the consequents of infinitely lar^e
order of the other arc AqKoCo, and for this reason we have not analyzed the
majority of the doubly asymptotic solutions.
Examples of Heteroclinous Solutions
403. Let us try to generalize, and let us set
F^-F<,-)-£F,.
Fq is a function of p, q and y, and F^ is a function of p, q, x and y.
These two functions are periodic, both in x and y.
Let us consider the curves
Fo = const.
(1)
388
in which we regard p as a parameter, and q and y are regarded as the coor-
dinates of a point.
Out of these curves, those which must draw our attention are the ones
having double points. These double points correspond to periodic solutions
of the canonical equations when we assume that e is zero and that F may be
reduced to Fq.
We have a double infinity of curves (1) whose general equation is
and which depend on two parameters p and h.
I have just stated that the most interesting ones are those which have
a double point, especially in the case in which some of these curves have
two or more double points. It is in this case that we shall encounter /398
heteroclinous solutions.
Just as in No. 225, let us try to formulate the function S of Jacobi,
and let us set
S =-- So + tSi-He'Sj-^-
The function Sq may be formulated immediately. We shall have
'/So rfSo
where q is a function of y defined by equation (1) and depending on two para-
meters p and h.
We then obtain
dp ~dx '^"dq -dy-^^''-"- ^2)
dFg dFn
We regard p as a constant in -r — ' -r— and Fi , and we replace q by its
value obtained from equation (1). Equation (2) Is therefore a linear equa-
tion with respect to the derivatives of Sj, whose coefficients are the given
functions of x and y, which depend in addition on the parameters h and p.
Since F^ ig periodic in x, I shall set
F, ^: J:'I',„e'"'^,
where ^^ only depends on y, just like the derivatives of Fq.
In the same way, I shall set
389
and the function li-m will be given by the equation
. ./P„ , di'o d.\,, (3)
dp dij dy
whose coefficients are the given functions of y.
This equation may clearly be integrated by quadratures.
Let us try to determine our asymptotic surfaces in this way. We must
first choose the constants h and p so that the curve (1) has a double point.
In addition, I shall assume that these constants are such that two real
values of q correspond to each value of y (this is what occurs in the
example presented in No. 225).
These two values of q are periodic functions of y, which become equal /399
to each other at the double point ~ for example, for y = yo-
Just as in No. 225, we may also assume that these two values of q are
the analytical extension of each other.
The function q then seems to us to be uniform in y and periodic of
y
period 4tt such as the function sin ^.
This uniform function will take the same value for y = yo and y = yo + 27t.
If we had several double points, instead of one, we could still regard
q as a uniform function of y of period 4tt, if the number of double points
were odd. On the other hand, if this number were even, we would have two
values for q which would not be interchanged when y increased by 2tt and
which could consequently be regarded as two different uniform functions of
y, having 2tt for the period.
In order to formulate our ideas more clearly, we shall assume that we
have two double points corresponding to the values yo and yi of y.
As a result, for y = yo and for y = yi, equation (1) must have a double
root, since the two values of q coincide, and consequently -^ must vanish.
Equation (3) is a linear equation with a second term, whose Integra-
tion is similar to the integration of an equation without a second term, and
consequently similar to the integration of the following equation
390
i^'Loo.,,f9.f =0 (4)
dp dq dy
from which we have
The function 9 thus defined is a holomorphic function of y for all
real values of this variable, except for the values y = yo, y = yi, which
correspond to the double points. For these values, the function 6 — which
plays a role similar to that of t = tan J in No. 226 — becomes zero or
infinite.
We then obtain /400
J iip
where C^ is an integration constant, from which we have
J dp
In order to obtain equations of asymptotic surfaces, we may write
assigning suitable values to the integration constants.
Let us first neglect e. We shall set S = Sq, and we shall assign the
values corresponding to the curve which has two double points to the con-
stants h and p = pg.
With this approximation, the differential equations have the following
as periodic solutions
(6)
P'-P'>< q-iu r-=yu
where yo , qo; yi , qi are the coordinates of the two double points.
In order to represent our asymptotic surfaces, we may take a point
in four-dimensional space, whose coordinates are
391
(/? -1^ a)co5.T, (p--- a)-'
rro;
Nj; p^^po, g^-q", y>yy\;
Nti p-^po, q = q', y>y\\
It can be seen that the two surfaces N^ + N'^ and Ni^ + N^ coincide
with this degree of approximation, just like the two surfaces N2 + N'2 and
N3 + N'3 .
Let us proceed to the following approximation, and let us set /402
O ^=- OQ -T- £ 5| .
In order to define S^, we must choose the constants Cm.
For the layers Nj and N'l, we must choose these constants so that the
functions \^ have a regular behavior for q = q', y = yo- We need only refer
to the analysis given on page 466, Volume II, in order to understand that
this condition is sufficient for completely determining its constants. I
shall call Si,i the function Si which is thus determined.
For the layers N2 and N'2, we shall choose the Cm's so that the ^s
are regular for q = q", y = yo, and we shall call Si 2 the ftmction Si
which is thus determined.
For the layers N2 and N'2, we shall choose the C^'s so that the i^^'s
are regular for q - q", y - yi. For the layers Ni^ and N'^, the %^s must
be regular for q = q', y = yi. We shall designate the two functions Si
which are thus determined by Si ^3 and Si i^.
The equations of our four surfaces thus become
Ni-(-N;;
/'
dS,.,
p
r/S,.,
?-?^^ dy-
p ■■
^/'» + ^ dr '
rfS, 3
^ = ^-^-^ dy'
p-
= ^» + ' dx '
1-1^' dy-
(8)
However, we should note that the function Si,i, for example, has a regu-
lar behavior for y = yo , and has an irregular behavior for y = yi . As a
result, our equations cease to be valid, even as a first approximation,
after the value yi is exceeded.
In order to provide a better illustration of this, I shall confine
393
myself to the following remarks.
Let y' and y" be two values of y such that
Jo r»'o; N', et N', ; >'o> J >ri — »''•
N, et N4; jo-t-it>;'>ri; N', etN;; }'i>r>yo-
For example, if the surfaces Ni and N\ coincide, the intersection will
correspond ^o a heteroclinous , doubly asymptotic solution -^i^^^^^i^J^^^,
very close to the periodic solution (5) for t = -". and very close to the
periodic solution (6) for t = +".
In order to determine this intersection, let us compare the equations
of Ni and N^
dS, 1 '^Sn
and the intersection will clearly be given by
rf(S,.i — Si^) _„ (9)
dx
Si.i - Sm is a function of x and y. which may be developed in positive and
negative whole powers of
¥e".
The fact that it is a periodic function of x is important to us. It there-
^re has at least a maximum and a minimum. Equation (9) therefore has at
ieastl^o solutions, which means that there are at least two heteroclinous
solutions .
In the same way, it could be shown that there are two solutions corres-
ponding to thrinter;ections of the surfaces N, and N'2 two corresponding
?o the surfaces N2 and N'a. and two corresponding to the surfaces N3 and N^.
The preceding analysis does not yield the homoclinous solutions.
394
Vw^
404. For example, let us set
Fo=— p — o'-H2nsin'^^:^sin« -^~-^' ,
1 a
F,= ncosarsin(y— 7<,)sin(^— ^Xi).
The periodic solutions (5) and (6) toward which the heteroclinous solutions /404
strive for t = -«> and t = +«= are then
X ~^ t, p^-q^-0, y ya,
X ---. C, p^- q -. o, y -.7,.
It will be noted that, for y = 0, F may be reduced to -p - q^. Therefore, for
y = 0, the function F depends only on variables of the first series p and q,
and does not depend on variables of the second serjes x and y. The function
F therefore has the form considered in Nos. 13, 125, etc.
Nevertheless, we shall not be content with this example, which proves
that the canonical equations having the form considered in No. 13 can have
heteroclinous solutions.
The two solutions (5) and (6) both correspond to the same value of the
quantities -7- and -7^ — i.e. ,
dx ily
(!c ' (It
However, these quantities -rr» "TT are nothing else than the numbers
which were called n^ and n2 above.
Therefore, we find that doubly asjrmptotic solutions exist, which come
infinitely close to two different periodic solutions for t = -<» and t = -H».
However, these two periodic solutions correspond to the same values of the
numbers n^ and vli.
Therefore, I shall formulate another example, in which we shall deal
with equations having the same form as those presented up to No. 13, and
which have doubly asymptotic solutions coming arbitrarily close to two period-
ic solutions which are not only different, but correspond to different values
ni
of the ratio — .
n2
Unfortunately, I would like to show that these solutions exist for
values of y which are close to 1, but I still am not able to establish the
fact that they also exist for small values of y.
405. We shall take two pairs of conjugate variables
395
or /405
^1. yi'y ^». y*<
by setting
This change in variables does not alter the canonical form of the
equations. We shall set
F = F<,(i — |Ji)-HftF,.
We shall assume that Fq is a holomorphlc function of xi and X2, inde-
„2 1
pendent of yi and ya, and that for xi = _, X2 = y, we have
— = O, -J = M
dx, axi
1 a^ ,
We shall also assvtme that for xg = y, xi = -y we have
dxi ' dx,
I shall assume that a < 1 holds for the quantity a.
It follows from these hypotheses that, if we set y = 0, from which we
have F = Fq, our equations will have two special periodic solutions.
The first solution, which I shall call a, may be written
a' >
^i = — > ^j= -' ri = '. ^» = o.
Ji = ac05/; rn = as\ne, ?i=i, t,,=o.
The second solution, which I shall call a', may be written
ar, = - , ^« = — ' ri = o. n = '.
$1 = 1. T,i = o, {j=acost, T), = a sin t.
The first corresponds to ni = 1, n2 = 0, and the second corresponds to
m = 0, n2 = 1. These two periodic solutions do not correspond to the same
value of the ratio — .
n2
In order to define Fj , I shall set
5i = I — rcosco, 5j = i — rsinio,
396
assigning a value which is essentially positive to the variable r.
I shall then assume that (due to the fact that p is a positive, /406
very small quantity) we have the following for r > p
F,= -^
^ f-'-^l _ (r-i)' ^ 'H^ (1)
r'
where iJj(u)) is a function of to, which is regular for every real value of co,
periodic with the period 2it, and finally which vanishes with its derivative
for 0) = and for ui = ^.
Since the function (1) would be infinite for r = — i.e., for 5i =
E,2 = 1 — I shall assume that for r 4p, the function Fj takes on arbitrary
values, in such a way that it nevertheless remains finite and continuous,
as well as its derivatives of the two first orders.
It may be readily verified that for y = 1 — i.e., for F = Fi — our
equations still have two periodic solutions ct and a'. For the first of
these solutions, we have o) = 0, and for the second we have o) = -S-.
It may be iiranediately concluded that for every value of u . our equations
will have these two periodic solutions.
406. We shall now integrate our equations in the case of y = 1 (assuming
at least that r constantly remains > p) .
If we first assumed that e = 0, we would be dealing with the problem of
central forces, and the integration would be immediately possible, This is
hardly true in the general case.
The Jacobi method leads to the partial differential equation
where h is a constant. Let us set
where k is a second constant, and we shall have
The general solution of our equations is therefore /407
397
/rdr
C2)
/a> ^ r 2J!L -_ =. k-, (3)
where h' and k' are two new constants.
We shall obtain our two periodic solutions a and a' , assigning the
following particular values to the constants
j,j , —
■i
Let us assume that we would like to employ equation (2) to define r as
a function of h' + t. If we assign values which are close to zero and
4 to the constants k and h. r will then be a periodic function of t + h'.
We shall set
u ^ n{t + A'),
Where the number n is chosen in such a way that r is a P"^^^^^ J'^f '"J^f
u with the period 2.. This number n. which is a type of mean motion, will
naturally depend on the constants h and k.
In the same way. ^ will be a periodic function of u.
For k = 0, we simply have
,• r : I -': if ill COS u.
L07 We therefore have two periodic solutions a and a' which will be
represented by two closed curves, if we may regard the ^'s and the n's as
tScoordinates of a point in four-dimensional ^P^^ J^/^^J^^ °^ ^"'^
faces pass through each of these curves -" °"%^^l°Sf ^^ J° f^l HH \^e /408
rui^^-frcerrin^efdrfa^^^^^^^^^
e is neglected.
In order to obtain the equations of these surfaces, it is sufficient to
398
Ct2
assign the values zero and — to k and h. We thus have
-■(?! -O'ii -+-(!.- Oil = /-/a»—(7--:r7)T,