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NEW
VOLUME THE
HI.
METHODS
OF
INTEGRAL
SECOND
TYPE,
INVARIANTS, DOUBLY By
Translation
of "Les
CELESTIAL
H.
M6thodes
MECHANICS
PERIODIC
ASYMPTOTIC
SOLUTIONS SOLUTIONS
Poincar6
Nouvelles
de la M6canique
C_leste.
Tome III. Invariants int_graux. - Solutions p_riodiques deuxi_me genre. Solutions doublement asymptotiques." Dover
NATIONAL For
sale
by the
Publications,
AERONAUTICS Clearinghouse Springfield,
for Virginia
OF
New
York,
1957
AND SPACE ADMINISTRATION Federal 22151
Scientific -
CFSTI
and price
Technical $3.00
Information
du
TT
F-452
'
(i,
P_,EQEDING
PAGE
BLANK
TABLE
NOT
OF
CONTENTS
VOLUME
III
CHAPTER INTEGRAL
FILMED.
XXII
INVARIANTS
Pa_m_ i Steady Motion Definition of
of a Fluid ............................................. Integral Invariants ....................................
Relationships Between the Invariants and the Integrals ............... Relative Invariants .................................................. Relationship Between the Invariants and the Variational Equation ..... Transformation of Other Relationships Change General
the Invariants Between the
..................................... Invariants and the Integrals .........
in Variables .................................................. Remarks ......................................................
CHAPTER FORMATION
4 7 9 15 19 27 32 34
XXIII
OF
INVARIANTS
43 Use of the Last Multiplier ........................................... Equations of Dynamics ................................................ Integral Invariants and the Characteristic Exponents ................. Use of Kepler Variables .............................................. Remarks on the Invariant Given in No. 256 ............................ Case of the Reduced Problem ..........................................
CHAPTER USE
OF
45 50 65 69 71
XXIV
INTEGRAL
INVARIANTS
73 Test Procedures ...................................................... Relationship to a Jacobi Theorem ..................................... Application to the Two-Body Problem .................................. Application to Asymptotic Solutions ..................................
CHAPTER INTEGRAL
INVARIANTS
AND
81 83 88
XXV ASYMPTOTIC
SOLUTIONS
Return to the Method of Bohlin ....................................... Relationship with Integral Invariants ................................
iii
91 115
Page
Another Discussion Method ............................................ Quadratic Invariants ................................................. Case of the Restricted Problem ....................................... CHAPTER POISSON
120 130 134
XXVI
STABILITY
Different Definitions of Stability ................................... Motion of a Liquid ................................................... Probabilitites ....................................................... Extension of the Preceding Results ................................... Application to the Restricted Problem ................................
142 143 153 156 159
Application
167
to
the
Three-Body
Problem
................................
CHAPTER THEORY
OF
XXVII CONSEQUENTS
Theory of Consequents ................................................ Invariant Curves ..................................................... Extension of the Preceding Results ................................... Application to Equations of Dynamics ................................. Application to the Restricted Problem ................................
CHAPTER PERIODIC Periodic Solutions Case in Which Time Application to the
XXVI II
SOLUTIONS
OF
THE
SECO_
TYPE
of the Second Type ................................ Does Not Enter Explicitly ......................... Equations of Dynamics .............................
Solutions of the Second Type for Equations of Dynamics ............... Theorems Considering the Maxima ...................................... Existence of Solutions of the Second Type ............................ Remarks ............................................... ] .............. Special Cases ........................................................
CHAPTER DIFFERENT
FORMS
OF
THE
176 179 188 190 197
228 232 241 245 246
XXIX
PRINCIPLE
OF
LEAST
ACTION
Different Forms of the Principle of Least Action ..................... Kinetic Focus ........................................................
iv
203 209 215
250 262
Page 268 270
Maupertuis Focus ..................................................... Application to Periodic Solutions .................................... Case of Stable Solutions ............................................. Unstable Solutions ................................................... CHAPTER FORMULATION Formulation of Solutions Effective Formulation of Discussion Discussion
OF
XXX
SOLUTIONS
of the Second the Solutions
OF
SECOND
TYPE
Type .......................... ...............................
to Equations
of No.
13
PROPERTIES of
the
Second
OF
Type
Stability and Instability Application to the Orbits
the
of
THE
SECOND
TYPE
of Least
Action
.......
............................................ of Darwin ..................................
PERIODIC Solutions
OF
Principle
CHAPTER
the
OF
Type
THE
SECOND
TYPE
................................
CHAPTER DOUBLY
329 340 348
XXXII
SOLUTIONS Second
322
XXXI
SOLUTIONS and
294 296 310 320
...................................
CHAPTER
Periodic
THE
........................................................... of Particular Cases .......................................
Application
Solutions
271 273
357
XXXIII
ASYMPTOTIC
SOLUTIONS
Different Methods of Geometric Representation ........................ Homoclinous Solutions ..................................... ........... Heteroclinous Solutions ..............................................
366 377 383
Comparison with No. 225 Examples of Heteroclinous
386 388
.............................................. Solutions ..................................
V
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NEW
METHODS
OF
CELESTIAL
VOLUME H.
MECHANICS
I II
POINCARE
ABSTRACT
Integral
invariants
are
introduced
using
the
steady motion of the fluid as an example. The usefulness of invariants in celestial mechanics is demonstrated. Various forms of the three-body problem are treated. Poisson stability is defined for the steady motion of a liquid, the general and restricted threebody problem. The theory of "consequents" is introduced in the discussion. The existence, stability, and properties of periodic solutions of the second type are treated. These are related to the principle of least action and the Darwin orbits. The concepts of kinetic focuses and Maupertius focuses are introduced in the discussion. Periodic solutions of the second type are treated. Homoclinous and heteroclinous doubly asymptotic solutions are discussed for the three-body problem.
CHAPTER INTEGRAL
Steady 233. of
In
integral
ticular Let velocity at time
order
to
clarify
invariants,
example us
from
consider
components t.
* Numbers
given
foreign
text.
in
it the an
of
the
XXII INVARIANTS
Motion the
of
origin
is useful
field
to
and start
importance with
of
a study
the of
idea
a par-
of physics.
arbitrary the molecule
margin
a Fluid
fluid,
and
which
indicate
has
let the
pagination
u, v,
w be
coordinates
in the
the x,
original
three y,
z
Wewill consider u, v, w as functions of t, x, y, z, and we will assumethat these functions are given. If u, v, w are independent of t and only depend on x, y, z, the motion of the fluid is said to be stead!. Wewill assumethat this condition is satisfied. The trajectory of an arbitrary molecule of the fluid a curve which is defined by the differential equation d__
dr _ d:
U
If
it
were
possible
to
is therefore (1)
g
integrate
these
equations,
one
would
obtain
/2
z = 71(6 xo, yo, zo), y --_,(t,,. xo, yo, Zo), z - ?3(t, Xo, Yo, Zo), such
that
their
x,
initial
If the position
the
figure
y
and
z would
values
x0,
be
Y0,
expressed
as
function
of
time
t and
z0.
initial position of a molecule of this same molecule at time
Let us consider fluid molecules F 0 at the initial instant of
placed, their group will form continuously deformed, and at consideration will form a new
a
were t.
the group time; when
known,
one
could
of which forms these molecules
deduce
a certain are dis-
a new figure which will move while being the time t the group of molecules under figure F.
We will assume that the movement of the fluid is continuous, i.e., u, v, w are continuous functions of x, y, z; there are therefore certain relationships between the figures F 0 and F which are obvious from the conditions of continuity.
will
be
If be
the figure a curve or
If the a simply
true
F 0 is a curve or a continuous a continuous surface.
figure F 0 is a simply connected volume.
If the figure F 0 is for the figure F.
In particular, is incompressible, able.
a curve
connected
or
let us examine i.e., where the
a closed
surface,
volume,
surface,
the
the
figure
figure
the
same
the case of liquids where volume of a mass of fluid
F
F will
will
hold
the fluid is invari-
Let us assumethat the figure F0 is a volume. At time t the mass of fluid which fills out this volume will occupy a different volume which will be nothing else than the figure F. The volume of the mass of fluid did not change; thus, F0 and F have the samevolume. Therefore, one can write (2) The first integral the volume F0. Wewill
is extended over the volume F and the other over /3
then say that the integral fffax
is
an
integral
by
the
It
invariant.
is known
that
the
condition
of
incompressibility
can
be
expressed
equation du dv d_ dx + _y + -37. =°"
The
two
equations
(2)
and
volume
Let us again consider of a mass of fluid
such
that
if
one
calls
p
(3) are
first
volume
F 0.
integral In
is
other
equivalent.
the case of is variable. the
density
f ff The
thus
ax
extended words,
(3)
of
a gas, Thus, the
i.e., the case where the the mass becomes invariable,
gas,
one
has
(4)
d,.=f f f podXo a.roa,.o. over
the
the
volume
F,
the
second
over
the
integral
fff is an
can
integral
invariant.
In this case, be written as
where
the motion
d(pu)
a_
is
d(po)
+--dT
steady,
the
equation
of
continuity
d(pw)
+
d_
=o.
(5)
The
conditions
234. second
The
us
hold
at
have
us
a a
for
assume
constant
of
(5)
remain
vortices
that
develop
us
Under
the
the
figure
that
the
In u,
these
Let
that
F
the
v,
this
that
satisfy
is the
that
are
of
along In
the
they
us
other
235.
words,
Due
presented
It
is
give
this
Let
that cases
simple
us
the
of
t
z
If
4
and
are
of
they their
could
it
is
with
a
The
same
compressible to
remains
or
forces
steady,
conditions.
It
is
not,
which it
is
not
useful
/4
the
integral
w dz).
it
has
is
an
the
same
to
value
along
the
curve
much
meaning
differential
functions
be
initial
can
be
the
of
= _
x,
x0,
examples
used in
of
by
which
which integral
invari-
generalizing it
is
not
of
the
their possible
form
dz = at, = -f
y,
I have
invariants.
equations
integrated, values
the
consideration
broader, to
invariant.
Invariants
question, the
invariants are
integral
Integral
the
one
these which
consider
of
lead
physical
given
curve.
subjected
motion
v dy+
integral
dx -_
Y,
closed
consider
shows,
nature
readily
for a
to
clear
definition
x,
us
F O.
Definition
to
provides
satisfied.
let
Helmholtz curve
a
only
certain here.
assumption,
is
whether
and
order w
F0
fluid,
conditions
assume
theorem
and
just ants.
Helmholtz
F.
(udz+ As
equivalent.
of
figure
temperature,
potential.
necessary to
assume
true
Let is
theory
and
example.
Let will
(4)
(i)
z.
one
would
Y0,
z0.
obtain
x,
y,
z
as
a
function
If we sent fine
the the
assume
that
the
coordinates of laws of motion
If these equations the point M at time
of
coordinates If group
x 0, Y0,
one
is represented
by
t and
M in space, point.
x,
y,
equations
are integrated once, one t, if its initial position
z0'
considers
of which
time
a moving point of this moving
z repre(i) de-
can find the position M 0, given by the
is known.
moving
points
which
obey
the
same
law
and
the
a figure F_ at the initial instant of time, the • different figure F at time t of these same points w!ll-form a will be a line, a surface, or a volume depending on whether the
group which figure
F 0 is Let
us
forms
a line, consider
a surface
or
a simple
a volume.
/5
integral (A dx + B dy + C dz),
(2)
where A, B, C are the known functions of x, y, and z. If F 0 is a line, it may happen that this integral (2) extended over all of the elements of the line F is a constant which is independent of time, and is consequently equal to the value of the elements of the line F 0. Let the
us
double
now
assume
that
this
F and
same
integral
F 0 are
A'
B'
C'
are
functions
integral has the same value the surface F, or over all Let triple
us now
assume
that
of
is
value
us
let
all
of
imagine
F and
(3)
dy),
and
z.
It may
extended over of the surface
F 0 are
volumes,
happen
all the F 0. and
let
that
this
elements
us
of
imagine
the
integral
a function for In
are
and
surfaces,
+
x, y,
which is of those
f f f
M
over
integral
dyd=+ B'a where
extended
F and
these
integral It
of x,
y,
z.
It
_l ax & d:;
is possible
(4)
that
it may
have
the
same
F 0.
different
cases,
we
say
that
that
the
simple
the
integrals
(2),
(3)
or
(4)
invariants.
occasionally
happens
integral
(2) will
only
have
the samevalue for the lines F and F0 if these two curves are closed, or the double integral (3) will only have the samevalue for the surfaces F and F0 if these two surfaces are closed. Wemay thus say that (2) is an integral to to
the the
closed closed
curves and surfaces.
that
(3) is
an
invarlant
with respect invariant with respect
integral
236. The geometric representation which we have employed plays no important role. We can thus lay it aside, and nothing prevents us from extending the preceding definitions to the case in which the number of variables is greater than three. Let us consider the following equations dxl
x_
where
XI,
one of
could t and
the
same
X2,
...,
their
the
.....
_
=at,
the
given
functions
them,
one
would
find
initial
terminology, and
dx_,
x,
X n are
integrate of
dx,
-
values
we
point
may
xp0,
point
..., M
the
x2,
...,
x_.
In order
system
of
x 0I, x0,
x n.
to
retain
values ...
x I, x2,
_.
Let us consider a group of points M 0 forming a subset F 0 and of corresponding points M forming another subset F (I).
Let
us
consider
an
integral
F are
of
the
continuous
order
subsets
(i) The
word
subset
is now
commonly
dxl,
...,
having
the
p
p
where _ is a function of Xl, x2, ..., Xn, and where of p differentials chosen among the n differentials dxl,
If
functions
group
F 0 and
of values
...,
x n as
Xn,
that n.
system
of Xl,
x I, x2,
...,
We shall assume dimensions where p _
M 0 the
x_,
call
(1)
d_
is
the
product
dxn.
employed,
so
that
I did
not
feel
it was necessary to recall the definition. Every continuous group of points (or system of values) is named this way: In threedimensional space, an arbitrary surface is a subset having two dimensions, and an arbitrary line is a subset having one dimension.
/--6
It
is
subsets
F It
two sets
possible and
may
to
F 0.
also
We happen
subsets F and are closed.
closed
give may
this thus
that
integral say
the
that
this
it
integral
same
is
an
has
F0, but only under the It is thus an integral
value
for
integral
the
condition invariant
the
two
invariant.
same
value
that with
these two subrespect to the
for
the
subsets.
Other
types
example,
let
lines.
It
of
us is
integral
assume
invariants
that
possible
p
that
=
may
1 and
the
that
the
same
may
also
be
value the
for
case
F
and
F0,
the
following
for
also
and
assumed.
F 0 may
be
For reduced
to
/___7
integral
f (A,dx,+ A, ax, +...+ has
be F
A, dx,,) = f _A,dx,
and
is
an
integral
invariant.
This
integral
IxBi dx_ + 2 X C,k dxi dxk,
where
B
and
C
I stated,
it
F
and
and
F0, The
are is
like
other
quantity
p
Let
us
A
of
that
the
will
called
Between
again
dr,
one
If the
could
integrate
integration
following
it,
were
all
the
dr.
of
performed,
may be
Xl,
x2,
...,
x n.
have
the
same
value
readily
order
of
Invariants
the
x, - _
If
may
the
consider
of
integral
examples
be
functions
this
similar
Relationships 237.
the
possible
the
and
As for
envisaged. integral
the
invariant.
Integrals
system dw,,
..... its
_
= at.
integral
the
result
(1) invariants
could
be
could
presented
be
formed.
in
form Yt = C,,
y, = c,, y,_ ,-= Cr,-t, "= t -_- Cn,
(2)
CI,
C2,
functions
...) of
C n are the
arbitrary
constants,
and
the
y's
and
z's
are
the
given
z for
the
new
variables,
x's.
Let us change instead of x's.
the
variables
by
taking
y's
and
Let us now consider an arbitrary integral invariant. Under the sign / (which will be repeated p times if the invariant is of the order p), this invariant must include a certain expression, the function of the /8 x's and of their differentials dx. After a change in the variables, this expression will become a function of the y's, z, and of their differentials dy and dz. Without changing the y's, in order to pass from one point of the figure F 0 to a corresponding point in the figure F, it is necessary to change z into z + t. Therefore_ when passing from an infinitely small arc of F 0 to the corresponding arc of F, the differentials dy and dz do not change (the quantity t which is added to z is, in effect, the same for the two ends of the arc). If one considers an infinitely small figure F 0 having an arbitrary number of dimensions F, the product of a number (equalling that
and the corresponding of the dimensions of
of differentials dy or dz will one figure to the other.
either
not
change
when
one
passes
In short, in order that an expression may be an integral it is necessary and sufficient that z is not contained in it; the dy's, and dz may be included in an arbitrary manner. Let discussed
us consider an expression in the preceding section
having
f
This
expression
represents
of xl, x2, ..., n differentials
Xn,
d_
is
an
We
would
like
to know
same
form
as
from
invariant, the y's,
that
which
_.Ad_o,
integral
a product
d.rl)
the
figure F 0 and F)
of
of
dx2,
whether
carrying out a change in variables expression (3) becomes fEB
the
order
p,
p differentials
. ..,
this as
(3)
a function
chosen
from
the
dxn.
is
an
indicated
d_',
A is
integral above,
invariant. we
find
that
By
we
B
is
tials
a
function chosen
of
the
from
y's
the
n
and
dy,, In
order
necessary only
that
and
depend Just
as
the
in
z,
_y,,
expression
sufficient
on
of
dm'
is
a
product
of
p
differen-
differentials
that
.",
dye-,,
dz. integral
(3)
be
an
all
of
the
B's
be
invariant,
it
is
independent
of
z
and
y's.
the
preceding
section,
let
us
again
consider
the
ex-
/9
pression
f
The
Bi's
and
After
the
the
Is
Ci. k
change
_x
B, dxf
are
in
+
functions
the
greater
symmetry
in r
the
xt-----yi, In
order
necessary
and
dependent
of
that
z,
and
that
depend
I have
(4)
be
an
of
the
on
238.
let
We to
us
are
the
now
led
closed
determine
the
to
Let
condition
by
A,dxj+
is
an
integral Let
and
our
us
invariant carry
integral
out will
with the
;
set
the
Xn=Z
B' i 's
following
,
integral
to
form
us
first
which
the
At dx2 +...+
respect
change
becomes
invariant, and
the
in
the
integral
assume simple
that
is be
in-
invariants p
=
i,
(i)
Andxn)
to
closed
variables
Bn-tdyn-t+
and
integral
lines. as
indicated
become BtJ/lq-B,d)',+..._
it
C'i.k'S
Invariants
attempt
subsets.
expression
y.
Relative
relative
x's.
t
n--I);
all
only
the
-t- "z_ C_.k dxidxk
(/----- T, 2.....
expression
(4)
this
notation,
.
sufficient
of
variables,
_/Z B_ dx?
For
_Ci.kdx_dxk,
B4dz)
t
above,
which I can write again, taking end of the preceding section
the
most
symmetrical
notation
from
B+d:;.
fx
the
(i')
This simple integral, extended over a closed, one-dimensional subset -i.e., over a closed line -- may be transformed by the Stokes theorem /i0 into a double integral extended over a non-closed, two-dimensional subset -- i.e., over a non-closed surface. We then have
f-'B,.d_;.: However, integral
lute
We In closed
all
be
can
J
fX' ¢,uB,._ abe) _._ \ dx_
the integral of the second member of invariant, and not only with respect therefore
order that lines it is
independent
Similarly,
and
conclude
(I) be an necessary
of
the
(2)
dxl / dxl dx'k.
(2) must be an absoto the closed subsets.
following:
integral invariant with and sufficient that the dBl
dBa
d_-;,
d.r;
respect to the binomials
z.
more
generally,
let
SA d_
(3)
be an integral expression of the order p, having the same which were discussed in the preceding sections. We would whether this is an integral invariant with respect to the of the order p.
form as those like to know closed subsets
Let us assume that this integral is extended over an arbitrary closed subset of the order p. A theorem similar to that of Stokes states that it may be transformed into an integral of the order p + i, extended over an arbitrary subset, which may be closed or not closed, of the order p + 1. The transformed integral may be written
fez
One ternately
i0
k 4-- _dA dxh. doJ.
always takes the sign + if p is even, and the if p is odd. [For additional details, refer
(_) signs to my
+ and - alreport on
the
residuals
to
my.report
de
l'Ecole
of
double
integrals
contained
in
condition
which
integral
invariant
of
that
be
(4) 239. let
an
us
with us
Let
M 0 be
its
that
respect
change
it
M
be
its
Volume
Edition
expression
is
to
closed
VIII),
of
the
and
Journal
and
Yl,
Y_,
the
make
z
.-.,
The
by
Bk'S
subsets p
the
be
+
is,
/1_!
is
i.
preceding
that
an
section,
an
integral
our
change
in
variables.
yn-s,
point
...,
Bk
of
yn-l,
will
writing
of
to
variables).
corresponding
appear,
order
invariant,
(i')
new
y_,
closed
the
(i)
(3)
lines.
form
F0
for
to
of
a relative
the
sufficient
respect
consider it
of
coordinates.
I will
p with
invariant
(with
the
and
integral
yl,
be
Centenary
necessary
order
to
a point
coordinates
Let
the
again
assume
Let
is
absolute
Let us
invariant
be
Special
Mathematica,
Polytechnique.]
The
and
the
(Acta
and
z+t
be in
F
functions
the
of
following
the
y's
and
of
z,
but
form
Bk(z).
If
that
the
is,
line
the
F 0 is
closed,
we
an
exact
depend it
not
must If
derivative
then
have
expression z [Bk(_
is
will
differential only
on
the
be
reduced
to
we
assume
that
of
Bk(Z)
which y's a
with
and
+
t) -- B,(_)]
I set z,
equal
but
also
(3)
dx',
to on
The
dV. t.
In
function order
V will
that
t =
o,
constant. t
is
infinitely
respect
to
small z,
and
expression
if
we
(3)
call may
B'k(Z) be
reduced
the to
11
The
expression x Bk.(z) dx_,
(4)
is then an exact differential which I set equal to dU. The function which is thus defined will depend on the y's and z, but it will no longer depend on t. I shall again make z appear by writing U(z). It then happens that dV
f + t)dx'_=jdU(
f
-37 =j-B'_(z
arbitrary
function
of
.. + t)=
U(z
is an
z of will
However, U(z) may be regarded as the derivative with another function W(z) which is also dependent on the then have
t.
d d--iW(.
On the other hand, may finally conclude
since that
The quantity +(t) designates be assumed to be zero without generality. One
then
+ t) = U(.
V must
V = W(z
/12
4-t)+f(t),
f(t)
we
U
be
-+ t).
reduced
+ t)--W(_:)+
an arbitrary essentially
respect to y's, and we
to
a constant
for
t = o,
?(t), function of restricting
t only, and may the conditions of
finds d B_.(.)= _
C k is
independent
of
z, so
that
f and the integral
of
12
first integral is an absolute
the
W(z)
expression
dW + f
(i')
may
be
reduced
to
XCkdx'k,
is that of an exact integral invariant.
240. In a similar way let us a higher order than the first.
+ Cx..
discuss Let us
differential,
a relative assume that
and
the
invariant
second
which
is
EA d_ is
this
invariant
The
which,
after
the
is
variables,
will
become
be
zero,
whatever
may
(i)
[B(z + t)-- B(z)] d_'= J
be
the
closed
subset
of
order
p
over
which
It
must
therefore
similar an
and
to
exact Let
now
limited
a
of
It
a
subset
integral
the
value
that
a
total
V
of
conditions"
differential
of
the
which
is
which first
v
subset of
p
-
i
p
dimensions,
dimensions
which
will
not
serve
closed
as
the
(i),
which
if itis
same
boundary
the
integral
equals
an
is
extended
calculated v, (i)
one
only
integral
for
of
over
other
will
obtain
depends
the
the
on
order
p
subset
similar the the
-
V,
subsets same
value
boundary
will
not
V',
V",
--
i.e.,
is
product
extended of
This tion
on will
p
-
over
the
v.
i
subset
(2)
is
v.
Let
its
changes by
v
and
while
clearly us
a
where
C
is
function
consider
its
_/7 do_'=
vB'(=
dm"
a
designates
function of
t,
of which
derivative
an the
arbitrary
y's,
depends
with
z and in
respect
t.
addito
t.
have
dt
As
(2)
subset
1 differentials,
integral the
the
be etc.,
j which
/13
order
it.
However,
having
certain"integrability
stating
consider
by for
The zero.
satisfy
those
differential.
us
boundary
We
it
extended.
are is
in
integral
fx must
change
last
expression
t
into
changing
z
t - h
=
shows, or
everywhere
when, into
this at z +
+ t; &,,'.
derivative the
same
does time,
one
not
change
transforms
when V
one (or
v)
h.
13
It
can
be
concluded
that
J----/Y D(z)
is
a
The
function
of
J has
the
following
form
D(z-4-t)dJ--fED(z)dJ,
x,
y,
z.
integral x D(z)d,o"
(3)
is of the order p - i, but it may be readily transformed into an integral of the order p. It is sufficient to apply the transformation which, in section No. 238, allowed us to change from the integral (3) to the integral (4), and which is the opposite of that by which, in the present section, we changed from the integral (i) to the integral (2).
to
The integral (3), the integral of the
extended order p
over
the
subset
v,
is
therefore
fEE(z)dt_' extended
over
the
subset
It
2.
It may
Under
is
zero
for
be
this
and
the
reduced
are
this
closed
to an
assumption,
integrals
However,
terminology employed for simple integrals, we (4) is an exact differential integral. And,
every
s= f
we
will
XE(z+t)d_'--
extended
equation
may
it is valid This
14
means
for that
an
arbitrary
subset;
integral
over
of
lesser
order.
have
f
__E(z)d_,',
the
also
fx[ B(.-+ o- E(z + t)] and
(4)
V.
By analogy with the may say that the integral in effect: i.
equal
subset
be written
= subset
V. as
E(.>] V.
follows
/14
v
.f_-[B(z)-is
an
absolute
We
therefore
Every tial
integral
241. of
the
so
that
In
order
The
same one
+
is
is an
the
be
the
However,
There
is
a
exact
in
If
invariant
of
of
the
p
so
order
that
which
is
if also
a relative
3,
then
to
us
Between
again
at
In into obtains
may
form the
order
xi + the
_i,
the
invariant p.
invariants, and
to
construct
very
quickly.
question one
be
is
wishes invariant
also
transform to
which
zero.
transformed,
one
is it
/15
illusory;
to
an
obtains
exact
again,
an
in-
differential
one
obtains
a
the
dx_ --
X2
of
dxn
variational
Chapter
equations,
and
we
disregard
the
linear
equations
dXk
dXk
(1)
equations
as
they
were
IV.
these
=
the
-- dr.
Xn
form
cl_,k
and
system
__.,.--
to
of
Invariants Equation
corresponding
beginning
system
in
transform
the
consider
XI We
differen-
zero.
dx,
defined
it
abandoned
invariant
Variational
Let
absolute
absolute
integral
is
this
wishes
be
which
The
p
exact
invariant
apply
procedure
order
an
successively.
to
lead
Relationship
242.
....
have
would
of
an
to to
invariant
but
how
applied
integral.
I,
one
seen
+
the
sum
from
would
the +
the
invariant.
continue p
the
would
an
integral, result
2,
which
differential
is
have
be
to
procedure
transformation
variant
also
p +
which
we
deduced
may
in
result:
integral
tempted
case
case
following
invariant
238,
be
order
this
the
No.
procedure
of
the
absolute
1 may
could
invariants
this
an
Section p
at
integral
and
d_'
invariant.
arrive
relative
integral
E(z)]
in squares
equations of
"+
dXk
the
(i) _i's.
we
change
One
x i
thus
(2)
15
There between of
is
the
a close
integrals
equations
relationship
which
of
(2) and
equations
may
be
the
readily integral
perceived invariants
(i).
Let F(h,
h,
..-,
i,,)=
¢onst.,
be an arbitrary integral of equations (2). This will be a homogeneous function of the _'s, which depends on the x's in an arbitrary manner. I can always assume that this function F is homogeneous of degree 1 with respect I would only have to a homogeneous Let
to the increase
function
us now
of
consider
degree the f
which
is We
an
integral
should
first
$'s. Because, if this F to a suitable power
F(dxt,
is
an
infinitesimal
infinitesimal of the first The
of
note
the
be
Under reduced
quantity
quantities order with
simple
d.c2,
invariant
F(dx,, of
of the respect
integral
(3) is
the case, to obtain
i.
following
that
were not in order
expression ...,
system
(i).
quantity
dx, .... the
(3)
dx,,),
under
the
sign
/16
f
, dxn)
first
order,
since
first order, and that to the quantities. therefore
dxl,
dx2
.._
dxn are
F is homogeneous
finite.
this assumption, let us first assume that the figure to an infinitesimal line, whose extremities have the
F 0 may following
coordinates
The quently
integral will
(3) may
be
reduced
to a single
element,
and
conse-
equal
Due to the fact that this expression is an integral of equations (2), it will remain constant and will have the same value for the line F 0 and
16
for
the
line
F.
If the line F0, and consequently the line F, are finite, we may divide the line F0 into infinitesimal parts. The integral (3), extended over one of these infinitesimal parts of F0, will equal the integral (3) extended over the corresponding infinitesimal part of F. The integral extended over the entire line F0 will equal the integral extended over the entire line F. Therefore, the integral
(3) is an integral
invariant. q.e.d.
Conversely, let us assumethat (3) is an integral the first order, and
invariant
of
F(_,,_,,.., b,)
will
be
and
In for
an
integral
reality, the line
F 0 is reduced coordinates
of
the
equations
the integral F, whatever
to
an
(3) must be the same for the line these lines may be, and particularly
infinitesimal
element
x_ As we
have
seen,
the
(2).
and
integral
whose
ends
have
the
FG if
following
x_+b.
(3) may
be
reduced
to
/17
F(_,,$,...., $.). Since
the It
integral
is
is an
therefore
an
Thus, an integral invariant of the first 243. higher
than Let
tions
Let
us
(2).
the
us
now
integral
of
this
expression
equations
these
two
be
constant.
q.e.d.
of equations (2) corresponds order of equations (I), and
to each integral vice versa.
to what
the
invariants
of
an
order
correspond. two
particular
arbitrary
solutions
of
the
equa-
Let
{h, _, _;, h _;...... ..... _,
be
(4) must
(2).
determine
first
consider
invariant,
(4)
(5)
solutions.
17
The following
which
depend
No matter duced to In
on
functions may exist
the
xi's
what the constants other
,
the
eL'S,
and
the
two chosen solutions, which are independent
words,
the
function
d_
T!
s
at
the
same
these functions of time.
F will
dXk
_i
be
dX1,.
an
time.
may
integral
of
be
the
re-
system
dXk
dt - d_T _' + _
_' +"" + _
_"' (6)
d_ = -_ dX_., _"÷ _-i which
the
and
Let us F has
that
and
_i's
the
_i's
formulate the form
Aik'S
It may
the
are
then
a more
definite
of
stated
that J =rE
is an
integral
invariant
mal
Let us assume that parallelogram whose
t
O
=
xl,
gram
dX_,. _,,, ,, _; + "" + -d-_
satisfy.
functions be
_
of
the
the
hypothesis,
x's
the
and
let
us
assume
alone.
double
integral
_t_,l.rid_k
equations
(I).
the figure F 0 may be reduced to an infinitesicorners have the following coordinates at
x_ + _l, xi + _,,
xt + ¢t-4- $i.
The figure F will also be similar to whose corners will have the following
an infinitesimal coordinates at
parallelot = t
x_, _ + b, _i + [_, _i + _i+ _. have
and the and
18
The integral precisely
J the
will be following
reduced value
-- since, under the hypothesis, system (6) -- it will have the FO.
to
a
single
element
which
this expression is an same value for the two
will
integral figures
of F
/18
Let
us
now
assume
that
F and
F 0 are
two
finite
surfaces.
Let
us divide F 0 into infinitesimal parallelograms, to each of which an elementary parallelogram of F will correspond. The value of J is therefore the same for each element of F 0 and for the corresponding element of F. It is therefore the same even for the entire surface F 0 and
for
the
entire
surface
The
integral
J is
The
converse
of
preceding
F.
therefore
this
may
an
be
integral
proven
in
invariant, the
same
q.e.d.
way
as
in
the
section.
244.
The
theorem
is
obviously
general,
and
may
be
applied
to
invariants the third
of a high order than two. Let us present it for those of order. Let us consider three special solutions of the equal (2), _i, _i, _i" These three solutions must satisfy the system T!
tions
d_,k
_
dXk
7i = _
777_, ._" (7)
d_;+ X_ aXe, , d_,_
'_
dXk
-RF = _'=J_ If
the
system
(7)
includes
an
,
_"
integral
of
the
form
I
ZAi.k.t
(8)
_k _;, _ ],
1 where
the
A's
are
functions
of
the
x's,
the
triple
integral (9)
XAiktdxid_kdxt will
be
an
integral
invariant
of
Transformation 245.
With
the
invariants
variational equation, one may make it possible to transform If
one
knew
a
certain
the
of thus
equations
the
(i),
of
vice
versa.
Invariants
reduced
to
the
readily find several these invariants.
number
and
integral
integrals procedures
invariants
of
of
the
which
the
equa-
tions
19
d J-i
dt -- X_, one could tions
deduce
from
each
of
them
an
(1)
integral
of
the
variational
equa-
d_,. %,_dX, d--7=Z., ,-t_ $_" By
combining
integral the
these
of
different
equations
integrals,
(2),
from
(2)
we
which
one
the
case
will
may
obtain
deduce
a
new
a new
invariant
of
equations(l).
Let
us
commence
by
studying
of
first-order
invariants.
Let
be
a
certain
will
be
number
functions
Now
of of
integrals
the
q
integral The
(dx_.), f
pend the
invariants
functions
of
under
the
the
depend
on
the
on the xi's differentials
xi's
in
an
and
must
be
homogeneous
first
(i).
These
integrals
F, (dxi)
their
same
equations
order
.....
of
of
these
t
the
...,
dxi's.
However,
with
They
will
respect
deto
dxn,
first
order.
Then
will of
2O
be the
integrals first
order
of
the
with
equations respect
(i).
Fq(dx_)
manner.
dxi
rq(dx_),
differentials
arbitrary
and
f
f
sign
dxl,
they
equations
F,(dx,) .....
Ft (dxi),
will
of alone.
let
fFt be
xi's
tO
(2)
and
the
_i's.
will
be
homogeneous
and
/2O
-V
¸
Now
let O(_,, _,, ...,_p; F;,F,, ...,Fq)= O[_k, Ft],
be
a function
of
an arbitrary with respect
the
manner, to the
_'s but F's.
and
of
which
the
F's,
which
is homogeneous
depends and
of
on the
the
@'s
first
in
order
Then O['>,,FI(_)] will be
be
a new
integral
a homogeneous
of
function
the
equations
and
of
the
(2).
first
In
order
addition, with
this
respect
will
to
the
$i's. It
thus
results
that
J /O[¢k,
is an
integral The
same
invariants
by
For
invariant result
of
could
changing
the
the be
Fddx_)]
first
order
of
readily
achieved
invariables
of No.
the when
equations
(i).
transforming
the
237.
example,
and
will
be
integral
246.
The
invariants. same
calculation
may
be
applied
to
invarlants
of
a higher
order. Let
be
the
p integrals
of
the
equations
(i),
and
/21
let
21
...j
be the q integral functions
invariants
of the xi's
of the second
and the products
order.
The F's will be
of the differentials
dxi dx k.
They will products.
be homogeneous
and of the first
order with respect
to these
Then
will be integrals
of the system
(6).
If O[%,F_] is
an
arbitrary
function
of the first order with
of
the
_'s
respect
and
of
the
F's,
which
is
homogeneous
to the F's, the expression
will be an integral of the equations (6). It will be homogeneous addition, and of the first order with respect to the determinants
As a result,
in
the double integral o['P_,Ft(dx_dxk)]
will
be
the
247. means
cedure
an integral
of
example,
invariants possible to
second
order
of
equations
(1).
of the same order, we thus have other invariants of the same order.
of the same order obtain new invariants
are
known, the of a different
same proorder.
I
assume, which and homogeneous
is func-
let fF,(dxi)
be two integral invariants the most general case, that tions of the differentials
22
the
Knowing several invariants of combining them to obtain
When several makes it For
invariant
,
of F 1
dx i.
the and
f e,(dx,), first order. F 2 are linear
/22
v
The
expressions
will
be
and
these
homogeneous will
In
the
and
be
same
of
the
integrals
first
of
order
equations
be
integrals
As
a
respect
to
the
$i's,
way,
F,($_), will
with (2).
of
the
F,(_')
equations
(6).
result,
(i0) will
be
an
integral
Since
As
of
F 1 and
a
F2
result,
the
are
system
linear
expression
(6).
with
(i0),
respect
which
to
the
changes
$i's,
sign
we
when
will
one
have
exchanges
!
the
_i's We
and may
the T
the
_
's,
does
conclude
function
that
of
coefficients
integral
therefore Now
the
depend
invariant
deduced
two
order and to
change
this
when
one
expression
changes (I0)
is
_i a
into
linear
_i
+
_i'
and
determinants
on
the
of
from
x's
alone,
but
not
on
the
$'s
the
equations
and
the
integral the
are n
second
order
expression
of
(i)
may
(i0).
let
and F2
the
this
fFj
be
not
's.
An be
_i
thus
homogeneous
and
the
invariants second
linear
and
differentials
is
(dx,.),
of of
equations
the
homogeneous dxi,
:F2(dxidxk)
the
second
(I); order.
functions, second
with
the I
first shall
the respect
first to
is
of
the
assume
that
with
respect
the
n(n-l) z
first F I
pro-
ducts
23
dxt dx_.
The
/23
functions
F,(_,),F,(_-- h_D will
be
integrals
The
of
the
system
(6).
expression
. i t + F,(_i)F,($i_;__$k_ )
will
be
an
On
integral
the
linear
and
other
of
the
hand,
it
homogeneous
system
(7).
may
readily
with
be
respect
_
An
integral
Now
invariant
of
two
the
&
to
verified the
invariants
We
which
I
Then
the
it
will
be
_
third
order
may
thus
of
can
deduce
can
write
F,(dz_dx.),
the
from
as
f
be
deduced
from
it.
second
it
follows,
two
that
is,
F,(dx, dz.)
order
of
integrals
for
purposes
equations
of
(1).
equations
of
(6)
--
brevity.
expression
+ F,(_U)F,(_"T) + F,(_"T) F,(_') F,(_t)F,(_'_") + Ft(_'_')F,(_') -4F,(,_') F,(_'_') -iF,(_'_') F,(_"_)
24
that
determinants
let
f be
(ii)
(12)
will
be
an
equations
integral
to
In
the
of
the
equations
addition,
this
will
be
respect to the determinants the corresponding quantities I shall linear
with An
from
continue respect
integral
It
by
adding
the
following
a linear
and
homogeneous
that
products of
F I and
the
F 2 are
function
quantities
with _i and
homogeneous
/24
and
dxidx k.
the
fourth
order
could
thus
be
deduced
(12).
should
be
noted
zero
when
we
to
obtained
formed, with,, four,,,of _i , _i ' _i "
assume
the
invariant
expression
equal
to to
system
(7)
that
this
invariant
does
not
become
exactly
set F,= F,.
Expression
(12),
divided
by
2, may
be
then
reduced
_, (_')F, (_'_") ÷ F, (,_') F, (_'_ ') + F, (_')F, An invariant invariant of the
of the fourth second order.
be obtained by the the order 2p would number)
order may always be An invariant of the
same procedure. be obtained from
In
general,
More generally, it (2p being an
deduced from an sixth order would an invariant of arbitrary even
let fF,,
be
two
arbitrary
p,
and
the
invariants
second
is
of
the
of
fF, equations
order
with
respect
to
the
products
(i);
the
first
is
of
the
order
q.
I shall assume that F I and F 2 are first with respect to the products
second
(_T).
•
248.
the
to
linear and homogeneous of p differentials dx,
functions, and the
of q differentials.
Let
_?',
_?' .....
_F+_'
25
be p + q solutions of equations (2). the system of differential equations
a_,_j dt = ,_dXk$_ _
of by
(i,k=,,_
These
....
solutions
,n;S=,
_" ....
Then
the
(13)
when each product determinant formed
, _P'.
In the same way, let F'2 represent the quantity when each product of q differentials is replaced by determinant formed by means of the q solutions
_F+,,,
satisfy
, _, "", p+q).
Then let F'I be the quantity which F 1 becomes p differentials is replaced by the corresponding means of the p solutions
_",
will
_F+,_....
/25
which F 2 becomes the corresponding
, gpw,.
product F_F;
will
be
an
Under
integral this
of
system
assumption,
(13). let
us make
_'_, _" .... undergo
an
arbitrary
permutation.
the
p +
q letters
, _Y+q' The
product
F'1F' 2 will
become
F;F; and
this
will
still
be
an
integral
of
system
(13).
We shall give this product the sign +, if the permutation under consideration belongs to the alternate group -- i.e., if it may be reduced to an even number of permutations between two letters. On the other hand, we shall assign the product permutation does not belong to the alternate group be reduced to an odd number of permutations between In any
case,
the
expression +F_FI
26
the - sign, if the -- i.e., if it may two letters.
(14)
will
be an integral
of system (13).
Wehave (p + q)! possible permutations; we will therefore obtain (p + q)! expressions similar to (14). However, we shall only have
which will be different. This is due to the fact that expression (14) does not change when the p letters which are included in F" 1 are only interchanged among them, and, on the other hand, when the q letters which are included in F'2 are only interchanged amongthem. Let us now take the sum of all the expressions (14). Weshall have an integral of system (13). However, this integral will be linear and homogeneous with respect to determinants of the order p + q, which can be formed with the letters
An
invariant
of
the
order
p +
q of
equations
(i) may
thus
be
/26
de-
duced. If p = q and if F I is identical to F2, the invariant thus obtained be equal to zero if p is odd. However, this will no longer be the if p is even, as I explained at the end of the preceding section.
will case
Other
let
249. us now
Based trace
Relationships the
Between the Integrals
Invariants
and
on the knowledge of a certain number of invariants, the manner in which we may deduce one or several inte-
grals. I shall
first
assume
that
we
know
two
invariants
of
the
tn_-h order
and
where ratio
M and M' _-will
M'
are be
an
functions
of
the
x's.
integral
of
equations
It may
be
stated
that
the
(i).
27
Let
us
consider
the
variational
_P', be n arbitrary equations. These n tions, which as system s. Let _i (k).
A be
solutions
equations
_p' .....
which
are
determinant
linearly
independent
of
these
a system of differential equa(6) and (7), which I shall designate
formed
by means
of
the
n2's
letters
Then M_
will
let
_"_
solutions will satisfy is similar to systems
the
(2) and
be
integrals
of
system
S.
_I'_
and
The
same
will
also
hold
for
the
ratio
M' M and, must
two
since be an
this ratio only depends on integral of equations (i).
The
same
Let
us perform
integral
result
may the
invariants
be
obtained
change will
the
x's,
in another
in variables
and
not
on
the
_'s,
it
manner.
as was
done
/2__17 in No.
237.
Our
become
fMS
d3, dy, . . . dy._,
dz,
dy, dy_ ...
dz
and fM'J
dyn-,
J designates the Jacobian or the working determinant of ables xl, x2, ..., xn with respect to the new variables Yn-l, z. According
to No.
237,
MJ yl_
and
this
is an (i).
also
integral
holds of
for
the
equations
and
M'J
must
y2,
-..,
Y,_-I,
ratio (I),
M' _--. this
only
Since ratio
depend
every is
an
the old variYl, Y2, ---,
on
function integral
of the of
Yi'S
equations
q.e.d.
28
250.
This
For
procedure
example,
may
be
the
also
p
in
several
...,
/Fv(dx,)
linear
invariants
./F,(d.r,),
of
the
first
order.
the
Mi's
It of
Let
us
assume
that
the
differentials
we
have FI = 312F_ + 313F_+...+
and
ways.
let
fF,(d.ri),
be
varied
depend
may
be
equations
Let
only
stated
on
that
the
x's,
the
Mi's
MpFp,
and , if
not
on
p _n
+
].,will
be
dx.
integrals
(i).
Aik
be
the
coefficient
of
dx k
in
F i.
We
must
then
have
ALk = M: A_.k + M_ A3.k-_-...+ MpA p._. Let
us
perform
invariants
then
the
change
in
7l:',(dx',), If
we
also
variables
as
was
done
in
No.
Our
237.
become
:F;(dx',),
...,
f
F'Adx'O.
/2S
set F_ = E A_k dx_.,
we
must
have A',.k = .M: A'_k + M: A'3k+...+
We the
shall
Mi's,
then
provided
have
n
that
p =< n
linear +
equations,
on
The
z.
integrals 251.
same
to
No.
is
237,
therefore
of" equations Now
the
an
integral.
which
only
on
we
may
obtain
!
Aik
true
s depend for
the
Mi's,
that
the
y's,
and
is,
the
Mi's
not are
(I).
let F(x,, x2, ...,
be
from
i. !
According
.M_ A_,_.
It
is
apparent
[\'_telF dx,
xn)
that
-+ _dF
dx, +.
dF ) . " +--_x_xnaxe,
29
will
be
an
integral
One
may
Let
us
then
invariant formulate
consider
an
No.
In 237;
order our
The
B's
and
order. question:
invariant
dxl+A_dx2+.
D
of
C's
must
this, let become
depend
us
on
the
sign the (1)?
make
the
y's,
dU is an exact following form
the
first
order
+Amdxn)
that the term under the f be the relationship between the integrals of equations
If this expression therefore have the
must
i
first
following
integral
to determine invariant will
the
the
the
f(A and let us assume tial. What will differential and
of
but
is an integral
exact
change
in
not
differential,
on
of
differenthis exact
variables
of
z.
the
function
U
U ----U0 +=Ut; U 0 and
U 1 are
integrals
of
equation
(i).
We
will
will
have
then
/29
have
dU --_
If we
return
to
the
old
dU
variables dU
therefore
results
integral
of
U is 252.
30
an
equations
integral We
could
dU
x,+...+
dU
_
Xt÷..+
(i). Ul = o,
and
we
_,
x..
that
dU Xt+
an
xi,
x,+ _
d_-q is
U|.
a_J
7i- = _ It
=
of cite
equations numerous
a_J
"
If
_'_ this
X,,
expression
is zero,
we
have
U = U0, (1). examples
of
this
type.
I shall
only
present
one
example.
Let
us
Let
&
Let
us
invariant
be
the
_'
Let
invariant
of
the
of
chan_e
in
the
the
first
order
quadratic
variables
having
form
according
the
form
_.
to
No.
237,
and
our
become
be
J be to
an
discriminant
make
will
Let
respect
consider
the
the
the
discriminant
Jacobian
x' 's.
of
or
We
will
the
the
quadratic
working
form
determinant
_'.
of
the
x's
with
B' 's and
the
C' 's)
have ._' _ _J_,
The
quantity
integral
of
Now
After
let
the
A'
will
equations an
be
(like
the
an
(i).
invariant
change
obviously
in
the
of
the
tn_-h order
variables
according
be
to
No.
237,
it
becomes
/3O
MJdr_dx;...dz,], and
MJ
must
I
may
be
an
conclude
integral
from
of
this
equations
(1).
that
A' 51_ji' i.e., A M_
must
be
an
integral
of
equations
(i).
31
Change
in
Variables
253. When the variables x i are changed in an arbitrary manner, without affecting the variable t which represents time, it is only necessary to apply the customary rules for the variable change for single or multiple definite integrals to the integral invariants. This is the procedure we have already followed several times. However, encountered. cannot lead Let
when the variable t is changed, greater It would even appear a priori that this to any result.
us
consider
the
system d_
l
dt = _ Let
us
introduce
a new
difficulty is transformation
d,r,
= x_
variable
dX n
......
(i)
x.,"
t 1 defined
by
the
relationship
dt dt I -- Z; Z is
the
given
System
function
(i) must
of
Xl,
x2_
...,
become dr,
dz',
dr,,,
dr, = ?._. - zx, ..... Let
us
coordinates
assume of
that
a certain
If the motion representing time,
of at
x n.
the
initial
point
values
M 0 in
(2)
1C"
space
0 x?,= x2, having
0
i
g
B
_
the
Xn represent
n dimensions.
this point is defined by equations (i), with the time t = _ this point will move to M.
t
On the other hand, if the motion is defined by equations (2), with ti representing time, at the time t I _ T the point M 0 will move to M'. Let different
us now consider points M 0.
If the motion equations (i), at
the
32
If the motion figure F 0 will
a
and the the time
figure
F 0 occupied
at
deformation of this t = T it will become
is defined by equations become a new figure _
(2), which
the
figure a new at is
time
zero
by
are defined figure F.
the time different
by
tI = x from F.
/31
Not
only
will
_
be
different
from
F, but
longer coincide with one of the positions is different from the time t = T. It
thus
appears
of the problem, duced from the invariants
of
Let
that
we
have
profoundly
us make
by
changed
and we must not expect that invariants of (I). However, order
in general
occupied
F
the
it will at
a time
given
no which
quantities
the invariants of (2) may this is what occurs for
be
de-
n.
the
change
in variables
of No.
237.
System
(i) will
become
at and
system
a_,
ay,
0
0
dr,
0
must
a_
0
(1')
I
(2) dt, = dy,
We
&n-,
then
assume
that
d.r._,
0
d_
(2 ' )
0
Z is expressed
as
functions
of
the
y's
and
of
Z.
Let
us
then
set
with integration being performed with assumed to be constants), and starting may
depend
on
System
the
have
Then
which
(2) becomes &'
dy,
0
will
y's are origin
y's.
dti
and
respect to z (the with an arbitrary
the
same
&__,
0
form
as
0
(2")
dz, I
(i').
/32
let M_Idzl...i_n,
be
an
changed
invariant according
of
the
order
to No.
n of
237,
equations
(i).
When
the
variables
are
it becomes
MJdy, dy,...ayn_,d_;
33
J is be a
the Jacobian function of And
of the x's the y's.
with
respect
to
the y's
and
z; MJ must
then
_ls&,&,...dy__,_, will
be
an
invariant
of
equations /MJdYtdy, T
(2"); dy__,dz
...
will
be
an
invariant
of
equations
(2'),
will
be
an
invartant
of
equations
(2).
General 253'.
Let
us
consider
and
finally
Remarks
a system
of
differential
equations
dxi = X_dt, and
their
variational
the
Let us assume first order
(1)
equations
that
equations
(i)
include
an
integral
invariant
of
E Ai dxF Expression On
with
Y,Ai_ i will the
E being
other
an
be
hand,
arbitrary
an
integral
these
equations
infinitesimal
Let z,. = _dt)
34
of
equations
(2).
(2) will
have
constant.
the
solution
/33
V-
be
an
arbitrary
solution
of
equations
(i).
If
c is
a very
small
con-
stant x_= ?_(t + _)=
will
still
be
a
solution
of _i=
will
be
a solution
It
thus
of
+ _ d-F
equations
(1),
?i(t+¢)--?i(t)=
equations
results
_(t)
and
dxi _ --dS-= _Xt
(2).
that 2A,,_i = s_ A_.X_
must
be
a constant.
Therefore, Let of
the
us now
second
EAiX i is an
integral
assume
equations
that
of
equations
(i)
include
(i). an
integral
invariant
order
f f
E Aik dxi dxk.
Then 2 hi,(i,_i--
will be
be
an
deduced Let
integral by
us
of
changing
equations the
"_k,o v,
(2)
of
equations
is permissible,
because
_i's
into
_=
_Xl,
and
(2'),
which
may
_'i "
set
with _ being a constant. solution of (2').
This
_
=
eX i is
a
Then "2,',;k( $,,Xk-- X_-$k) will
be
an
integral
of
(2).
This
shows
that
35
is
an
integral
This order
n
invariant
procedure -
i,
may
sometimes
may
be
Let
one
now
the
it
knows
first
order
possible an
illusory,
to
us
makes
when be
equal
of
to
invariant
because
the
of
equations
obtain
an
of
order
the
invarlant
(i).
invariant n.
which
is
of The thus
the
/34
procedure obtained
zero.
envisage
an
invariant
having
the
following
form
Z(Ai+tBl)dzt, where
Ai
having
and
this
Bi
are
form
functions
of
the
We
x's.
shall
encounter
invariants
below.
Then Z(Ai+tBl)_t
will
be
an
must
be
a
integral
of
equations
(2).
As
a
result,
E(A_+tBi)Xl
For
constant.
purposes
of
brevity,
let
us
'P = ZAiX_.; and
the
set
q', = $BiXI,
expression "b + tap t
must
be
a
constant,
which
entails d'P )-7 +
the
dq't t o-7 +
condition
,l,, = o,
or
a,
_x_xlXt +q',+
same
36
The Xi's , the Ai's holds true for
, and
the
(3)
tX, a,b
z.. _-x7 Xi = o.
Bi's
are
functions
of
the
x's.
The
+, The
identity
(3)
'h,
can
A__X,.,
only
_Xt.
occur
E
if we
also
have
identically
d, l,I Xl = o 2_i
and _d,l' xi. + 'h = o.
The
first
relationship
253".
shows
us
that
_I is
an
integral
of
equations
(I).
Let q_
--=
COIISI.
be an integral of equations (2). The function _ must be of a specific form, a whole and homogeneous polynomial with respect to the _i's, where the coefficients depend on the xi's in an arbitrary manner. Let
m be
the
degree
of
this
polynomial.
The
(where _' is nothing else than _, where the $i's differentials dx i) will be an integral invariant Under specific Let equations
this form us
assumption,
let
I be
an
arbitrary
expression
were replaced of equations invariant
by the (i).
of
the
237,
and
_.
make
the
(I) will
change
in variables
according
to No.
the
become dr; _ TIT -o,
d; _- =,,
and, if one employs n i and _ to designate the variational equations of (I') will be
(i')
the variations reduced to
of Yi
and
z,
d_, d_ dt -- _ --=o.
37
/35
With
these
new
variables,
_ will
have
the
specific
form
_0, which
is whol_
homogeneous, and has the degree m with respect to the ni's and _. The coefficients may be arbitrary functions of the Yi'S. However, according to the theory presented in No. 237, since we are dealing with
an
integral
The xi's relationships
invariant,
these
cannot
are functions of the y's and of z, and between the variations may be deduced
_i _ X
The_'sare
coefficients
therefore
linear
clz¢
functions
One then passes from the (4), whose determinant
Let I of _.
I 0 be the invariant We will have
of
form _ to is J. of
the
on
z.
following
(4)
d_ i
the
n's
and
terminant of these linear equations (4) is nothing bian of the x's with respect to y and to z. I have J.
tution
depend
_0,
which
the
form
of
_, and
the
de-
else than the Jacocalled the Jacobian
_0 by
corresponds
linear
to
the
substi-
invariant
I = loJP with
p being
the
degree
of
the
invariant.
However, I 0 is a function of the coefficients of _0 and, consequently, a function of the y's, which is independent of z. It is therefore an integral of equations (i). Let we have
and
M be
the
last
multiplier
of
an
(i).
38
(i),
in
such
a way
that
that 31 dx t dz, ...
is
equations
integral
invariant
We have seen Therefore,
in No.
of 252
the
order
that
dxn n.
MJ will
be
an
integral
of
equations
/36
Iu(MJ)p-=
I_Ip
will be an integral of equations (i). therefore corresponds to each invariant Now
let
C be
respect to the variables _. If
C O is
a covariant
coefficients
the
of of
the
_,
corresponding
An integral of these of the form _.
form
and
the
covariant
_, having degree
of
the
q with
_0, we
equations
degree
p with
respect
will
to
the
have
C =-- CoJP.
The and they those of
coefficients are therefore
Therefore,
is an
C_[p is
integral
C, where We
the
of C O are independent
an
integral
invarlant _i's
therefore
functions of z.
of
of
been
have
a method
replaced of
invariants. the so-called
The particular case absolute invariants
attention.
If
will form
be an a new The
order.
integral integral
same For
C,
for
example,
example,
may
(i), by
(2);
true
of for
therefore,
where
C'
is none
other
than
forming
an
applied
/37
dx i. a great
number
of
integral
absolute
covariant
to
integral
of
of
of equations (i). One may therefore without knowing the last multiplier be
_0,
in which p is zero (i.e., the case or covariants) merits particular is
invariant invariant
procedure
equations
equations
have
of the coefficients The same holds
invariants
M.
of higher
let EA,-_dxid_rk
be
an
integral
invariant
of
the
second
order.
The
bilinear
form
39
which is integral
an integral invariant.
of
equations
(2)
and
(2')
is connected
with
Every invariant or covariant of this form, multiplied by appropriate power of M, will be an integral of equations (2), and will consequently produce a new integral invariant.
this
one (2')
In the same way, if one has a system of integral invariants, a system of forms which are similar to _ may be deduced from it, which will be integrals of equations (2), (2'). An integral of equations (i) will correspond to every invariant of this system of forms. An integral invariant of equations (i) will correspond to every covariant of this system of forms. For
example,
let
F and
F 1 be
two
quadratic
forms
with
respect
the _'s. They become F' and _i when the _i's are replaced ferentials dx i. Let us assume that F and F I are integrals that, consequently,
are
integral Let
us
invariants consider
the
of
by of
to
the dif(2) and
(i).
form
/38 F--_F,
where % is an unknown. When stating that the discriminant of this form is zero, we shall obtain an algebraic equation of degree n in %, for which the n roots will obviously be absolute invariants of the system of forms F, F I. These will therefore be integrals of equations (i).
and
However, this is not all. F and F I can be written in
Let %1, the form
12,
''',
In be
these
roots,
F ----l,h_+itA|+...+XnA_, F,= AT+ A|+...+ At, with AI, A2, ..., A n being the purely algebraic operations. The
40
quantities
AI,
A2,
linear
...,
forms
A n may
be
which
regarded
may
as
be
determined
the
covariants
by
of zero degree of the F, FI system, so that
are the integral invariants of equations (i), if A_ designates what Ai becomeswhen the _i's are replaced by the differentials dxi. However, there would be an exception if the equation for X had multiple roots. For example, if XI were equal to %2, it could no longer be stated that
are integral
invariants,
is an integral
but only that
invariant.
Nowlet f
be
two
integral
EAikdxidxh,
invariants
f
of
ZBikdxidxk
the
second
The
order.
two
bilinear
/39
forms
¢,, = z B,k(_A_,-- _*-_') will
let
be
The most n = 2m. Let
and
integrals
let
equation
us
(2) and
interesting
consider
us make for
of
its
X of
the
(2').
case
is
in which
n
is even;
therefore,
form
determinant
degree
that
n =
equal 2m.
to
However,
0.
We the
shall first
have term
an
algebraic
in this
41
equation tion of
is a perfect order m. The
will
integrals
be Now
_ and
square, m roots
of
_I
so
equations
can
that
(i),
for
the
in
the
form
be written
,I, = Z
it may
ti(PiQ_--
+, = 2(P,Q;-
the
be covariants (2') to which
There would multiple roots. If we
to
an equa-
reason
as
above.
QiP_) polynomials polynomials,
with respect where the
to the _m'S
expressions I',Q_--QIP_,
wii1 (2),
same
reduced
Q,P;)
and the Pi's and the Qi's are 2m linear _'s. The P_'s and the Q_'sare the same have been replaced by the _''s . Then
be
of the be
have,
the system integral
an
for
P2Q_--Q_P_
....
,
PmQ_-Q,_P_
0, 01, invariants
exception
to
and
this
consequently will correspond.
if
the
equation
integrals
for
of
X had
example, it = t,
it
could
no
longer
be
stated P,Q;-
are
integrals
of
(2),
(2'),
that
the
PiQ,,
P,Q;-
but
only
two
that
P, Qi -- Pi Q, + P,Q; is
42
an
integral
of
(2),
(2').
expressions
P;Q, their P;Q,
sum
/4O
CHAPTER XXIII. FORMATION OF INVARIANTS
Use 254. readily
There
when
is
the
an
last
of
the
integral
Last
Multiplier
invariant
multiplier
of
the
which
may
differential
be
formed
very
equations
is
/41 known.
Let dxl X,
be
our
differential
Let that
we
us
that
have,
there
function It may
M
is
then
dt,
(1)
is a function
M
of Xl,
X2_
...,
order
n
x n,
so
identically d(MX,) dxi
This
dx_ _-=
equations.
assume
also
dx2 =-XV .....
be
d(_X,) dx2
called
the
stated
d(MX.) dxn
last
that
_0.
multiplier.
the
integral
of
the
J =f_ld_,dx_...dxn is an
integral
integrated; integration
the
integral
invariant.
expressing constants
J will
Let
x I, x2,
assume
...,
xn
that as
equations
functions
of
(i)
have
t and
of
n
been
of
the
become J =fM .I
The quantity with respect
us
Adatd_... d_,,
A is the Jacobian or to the _'s. We will
the functional then have
determinant
x's /42
dJ f dM dt =.] --d[- d_l d_2 "" d_"
43
However, dM t_ = M d._ + ', dM d--7-_5 -_; dM
On
the
other
_ "--
hand,
I may only write the first deduced from it by changing Therefore,
A +
dt
dA _
dM
dxt d.v, -_=1' c-_t''"'
line _i
of this to _2,
must
be
xi-+dt_ with
respect
to
the
a's.
the xi's with respect to x i + Xidt's with respect
This
dx,! . d=t
determinant; _3, .-., an.
the
Jacobian
of
the
others
may
be
the
=x,.+Xidt will
be
the
product
of
the
the _'s -- i.e., of _ and the to the xi' % which I shall call
Jacobian
of
Jacobian of the D. I may write
dA A+dt-_=a.D. However, the Jacobian D may be readily formed. The elements principal diagonal are finite, and that belonging to the _ to the _ column may be written
The other elements line and to the _
are infinitely small; that belonging column (i < > k) may be written
to
of the line and
the
dXt d;ck
will
It thus have
results
that,
neglecting
"
terms
on
the
order
of dt 2, we
/
dXi D = r+ dt'_ z.a ,' from
44
which
it follows
that
/43
d-_
dX_
_ One
may
conclude
which
we
,T_,"
that
dM dt
from
- _
dM
finally
dX,.
E
d(MX_)
have
dJ dt
_0,
q.e.d.
Equations 255.
In
the
gral invariants learned how to equations, gral
and
in
invariants The
case
may form
the
integral
preceding
integral
is
of
the
I obtained
second
which a
will
second
dynamics,
we
3,
Vol.
order
integral and
which
order
may
and
integral
be
A
be
written
number
56 the how
of
inte-
on, we variational to
deduce
inte-
d_+_Z,+...+
d_.r.).
of
the
little
which
I may
as
follows
it
greatest farther
I may
deduce
167)
is
as
follows
const. from
which
j,=fz
p.
deduced
is
follow.
invariant may
great
learned
I,
_i _, + _ _, - _i _, + ....
is
as
follows
Importance on
(still
it
is
for p.
the
167,
Vol.
I),
write
=
The
a
From Sections integrals of
chapter
(equations
invariant
statements
of
formed. number of
J,=fcd_,dz,+ It
Dynamics
them.
_ _,The
equations
be readily a certain
from
first
of
of
CONSt.
from
of
the
fourth
dxtdyidxkdyk.
45
The
summation
combinations In
where
of
the
the
indicated the
same
by
the
indices
way,
summation
the
is
sign
i and
7
may
be
extended
over
the
n(n
- i) 2
/44
k.
integral
extended
over
n(n
-
i)
(n - 2)
combinations
of
6 the
three
indices
i,
k and
i, will
still
be
an
invariant,
and
so
on.
We thus obtain n integral invariants if we have n pairs of conjugate variables. One of these invariants Jl will be of the second order; another J2 will be of the fourth order; another J3 will be of the sixth order,
...,
and
However, different. the second an
it
last
is not
Jn will
be
necessary
of
to
the
assume
order that
2n. these
invariants
are
At the end of No. 247, I stated that from an invariant order, one can always deduce an invariant of the fourth
invariant
Jn which from the
the
of
the
sixth
order,
I have just defined first of them.
These
invariants
may
be
and
are
so
none
on.
The
other
than
considered
in
invariants those
another
which
way.
At
Jl,
J2,
may
the
be
all
of order, --., deduced
beginning
of page 169, Volume I, I demonstrated the manner in which one could deduce the Poisson theorem from the integral (3) on page 157, or -- which amounts
to
the
same
thing
--
from
the
Following the same procedure with a theorem similar to that of Poisson.
integral
the
invariant
invariant
J2,
Let
be
four
integrals
of
the
equations
of
dynamics.
Let
be
the
Jacobian
of
these
four
integrals
'z'i,
46
yl,
Xk,
with
yk.
respect
to
J1-
one
would
obtain
The
expression Y'-,_Ik,
where i,k,
the will A
the
summation still
similar
invariants
is
be
an
extended
over
all
combinations
of
the
indices
integral.
theorem
would
J3,
"'',
Jq,
be
obtained
by
commencing
with
any
of
Jn"
of
However, according to the statements which I have just made, the theorems is different from that of Poisson in reality.
be
However, attributed
It It
could be is known
from among all to the last of
obtained that the
of these them
invariants,
great
importance
by the procedure given in the preceding equations of dynamics have unity as the
none
may
section. last
multiplier. 256. coordinates on
page On
I shall now assume that the x's designate of n points in space, and I shall employ
169 page
of Vol. 170,
the the
rectangular notation given
i.
we
obtained
the
following
integral
of
the
variational
equations NfyT_
The
corresponding
,_7 dV
integral
invariant
\', [ v ,It
In
the
same
way,
the
to the
written
dV dx).
+...
+ &,,,)
integral
_'r_l
The
be
invariant
( &,, + &,, corresponds
may
i --_ const.
invariant
_ (xli
d)" _.i --.)"
li dx2i
--
x2i
rlyli
+
y2i
dxli)
47
_J
corresponds
to
the
integral z,(x,i_,,i-)q,_i--
However,
none
of
these
invariants
may be immediately deduced of gravity, and area. This does not hold function V is homogeneous In No. variational
_,,i_,. + y._,_,,-) -- const.
from
is the
of
56, we learned equations have
removing
the
It may the same
indices,
be stated procedure
we
mi
which
we
invariant
obtain
The
which
second
the
has
integral
48
an
the
degree
-i,
a very
integral
the
-4- const.,
E #' \ m
dx
_
be
order
invariant
special
dxdVdx) '
nature
since
it
depends
on
time.
be
readily
written
\ :znz
of
of
+c°nst'
--#Ydx)-+(P--2)t/EO_Y
may
/4_k
dV _) + consl. d,r
\ m
integral
f and is t_erefore seen that
_ki
when
more generally that if V is homogeneous leads to the following integral
J--/E(2xd)
an
dxhi
of
they
center
have
E(_txvj--py_)=(a--p)t
from
since
energy,
that if V is homogeneous the integral
E(2x,,+y_)=3tZ(y__
p,
interest, of
for the following, which occurs with respect to the x's.
\
or,
great
integrals
an
._,,),
exact
differential.
It may
is
none
other
The gral the
the
invariant
taken values
arc
chapter. as
As
J
energy
is
of
constant,
the
first
is
the
figure
When
this
I
explained
a result,
which
figure
is
in
the
we
The
we
therefore
of
a
to
time.
not
curve
is
integral
single
_" --py
is
value
This it
in
to
the
this
if
figure
is is
one
if
inte-
C1 be arc.
F 0
in
the
preceding
become
F,
C O
and
C 1 do
not
4_7
Co).
the
two
is
reduced
variations
ends
of
to are
the
arc
an
arc
proportional
correspond
true
if a
the
arc
relative
the
curve
invariant,
of
as
is I
desig-
chapter. that
the
may
the be
integral.
constant
arc
of
the
added
under
For
example,
curve the we
f
is
closed,
sign
without
may
an
add
_),
coefficient.
integrals f
relative
(which these
constant.
also
assumes
of
dx)
F
the
therefore
differential
value
arbitrary
Thus,
also
an
Let C O and ends of this
_)t(CI--
.,y
_(zdy+y
are
therefore
called
dx)-}-(p--
_._..e,:_....
energy
preceding
exact
changing
the
integral
However, arbitrary
an
C.
chapter.
however,
constant,
for
the
when
deformed;
particular,
closed.
with
is
call
have
constant
here)
The
nated
have
shall
integral
is
In
it
curve. the two
preceding
.f:,:
a
I
order;
deformed
J =fE(2x ,/
to
which
along an arc of an arbitrary for the energy constant at
This
change,
than
Z y dx,
fear
dy
invariants.
49
We may The
saw
in No.
always be invariant
238
deduced of the
that
an
absolute
invariant
from a relative invariant second order which is thus
of
the
second
order
of the first order. obtained is none other
than Jr_=/V dx dy, which
we
studied
This
is
above.
the
case
in which
the
expression
which appears under the f sign, becomes an exact differential. This is the case in which p = -2, which would occur if the attraction, instead of following Newton's law, follov_ed the inverse of the cube of the distance. We then have X(_x_ The
quantity
respect
Z2xy
to
is
time,
and
therefore
--2ydx)=
E2xy.
a polynomial dx
Z2xy=
the to
expression time.
Zmx 2
Jacobi
reached
However,
in
of
the
first
degree
with
since
is
a polynomial
this
result
d = _ Emx_,
E_mx-_
of
at
the
the
second
beginning
degree
of
with
his
respect
Vorlesungen.*
general, X(2xd:--pydx)
is not In the
an
exact
the
differential.
special
following
case
of Newtonian
fX(2xdy+y
Integral 257. invariants *
50
attraction,
Invariants
dx)--3t(C,--
and
invariant
takes
Note:
English
Co).
Characteristic
It may be asked whether there in addition to those which we
Translator's
our
form
title
is
are other have just Lectures.
Exponents algebraic formed.
integral
/48
Either Chapters
IV
invariants and
the
the
method
and
V,
to
least
with
Let
may
respect
us
set
an
method
As
be
we
which
have
of
the
applied
motion
more
to
the
integrals
of
be
or
employed. the
could
equations it
Bruns,
be to
procedures the
However, at
may
correspond same
applied
of
to
I
seen,
employed the
integral
variational these
in
equations,
equations
as
are
themselves.
advantageous
to
modify
these
procedures,
form.
arbitrary
system
of
differential
equations
d'_ - Xt, dt
and
their
variational
/49
equations (2)
d._ dXl dt ----'X
Let order
us
first
having
try
the
to
determine
which
the
expression
differentials
dx,
These
_k.
the
integral
invariants
of
first
form (BL dzt
in
(1)
and
invariants
+ B, dx, +...+
under where
the the
sign
B's
correspond
B_ dz.),
to
f
is
(3)
linear
are
algebraic
the
linear
with
respect
functions
integrals
to
of
of
the
the
x's.
equations
(2). What which the
are
are
a
us
assume
periodic
will
be
period the
conditions with
under
respect
to
which the
equations
_'s
and
(2)
algebraic
have
integrals
with
respect
to
x's? Let
to
the
linear
the T.
that
solution known One
following
values of
functions
can
then
are
period of
derive
assigned
T. t,
The
which
the
to
the
x's
which
coefficients will
general
be
of
periodic
solution
of
correspond
equations and
have
equations
(2) the (2)
form
_ = X_Ak_,'+_,_ .
The
_i,k'S
ponents, We
are and
can
in
periodic the
then
Ak'S solve
functions are the
integration linear
of
t,
the
(4)
_k'S
are
characteristic
ex-
constants. equations
(4)
with
respect
to
the
51
unknowns
Ake_k t, and
we
will
obtain Ake _d =
are
periodic
functions
El_tBt,k
The
0i,k'S
of
the
There will therefore be n relationships _'s, and there will be no others.
(5)
t. of
the
form
(5) between
If equations (i) and (2) include q different integrals which are linear with respect to the _'s and algebraic with respect to the x's, some of these q integrals may cease to be different when the x's are replaced by of equations What
the values (i).
may
then
be
corresponding
to
one
of
the
periodic
solutions
done?
/5O
Let
1Ii=
be of
these q linear the x's, which
Bli_,-_-.B_i_t-+...+Bni_=
eonst.
(i=r,2,...,q)
integrals, where the B's will will correspond to q integral
be algebraic functions invariants of the form
(3). They are different between them having the
-- i.e., following
there form
are
no
identical
relationships
_, IT,+ O,IT,+. + 3_"_ = o, where
the
coefficients
8 are
constants.
Neither
(6) does
the
following
form
Occur
+, lit +
with
the Is
having
_'s
it the
being
integrals
then possible that following form
'_t lit +...
of
+ +y llq =
equations
there
may
?t lit-i-?Ill2 +...+
(6')
o,
(i). be
a relationship
?yII V_-=_ o,
between
them
(6")
with the @'s being arbitrary functions of the x's alone. According No. 250, if the same relationships hold, the ratios of the functions must be the integrals of equations (I).
52
to
Wewill
therefore have ?, ",
with
the
_'s
being
integrals,
'?_. +2 and
_ ?_ _+I
consequently
,.., If, -I- "+++,_ TI2+. •. -i- '5,I IT,,,_ o, which
is An
between
contrary
to
identity the
our
hypothesis.
relationship
of
the
form
(6")
cannot
therefore
exist
Hi's.
However, if the values corresponding to one special solution, whether it is periodic or not, are assigned to the x's, it could happen that the first term of (6) vanishes identically. It could happen even if equation (6) which is not identically satisfied whatever may be the /51 )
x's, would hold when functions of t, that special
shall
the x's are replaced by the appropriately is, by those functions which correspond
to
chosen a
solution.
Every special designate as Under The
this case
solution under which a singular solution.
assumption,
in which
the
two
cases
this
may
phenomenon
be
periodic
solutions
in which
they
not
consider
a singular
is
produced,
I
presented. of
equations
(I)
are
all
singular; Or,
the
case
258.
Let
which
it
us
are
_,Bk.,+ from
singular.
solution
_,B_., +...+
$.
Let
us
set
_qBk.q = B+,
follows B,_+
Since
all
relationship
(6) was
B,_, +...+
not
B,,_,, = _,I;_+
identically
_,H, +...+
verified,
_qHq.
we
do not
have
iden-
tically
R,= D, .....
B.=o.
(7)
However, since relationship (6) must be verified by the solution S, these relationships (7) (which are algebraic, according to our hypotheses) must be satisfied for the values of the x's which correspond to the
53
solution S. Nowlet us set dBl
_= x, _
and thus
dBl
dBt
+x, 7_, +'"+
x_ _-_x.'
dB_
aB:.+
B_=Xt-_t+X2dx
The
solution
S must
_
obviously
• • "+
Xn
satisfy
dB}
d--_x_ .....
the
relationships
(7') and
the
and
so
B_=o
(i=,,=
....
,n),
B_=o
(i= ,,_, ...,n)
relationships (7")
on.
We shall therefore successively form the relationships (7), (7'), (7"), etc., and we shall stop when we have arrived at a system of relationships which will only be the result of those which will have been previously formed. Relationships (7), (7'), (7"), etc., will be algebraic to our hypotheses, and all of them together will form what called in No. ii a system of invariant relationships. Therefore, if a system of differential periodic solution, it will permit a system tionships. It
is probable
that
the
three-body
braic invariant relationships except am still not able to prove this. Under solutions.
this For
equations permits a singular, of algebraic invariant rela-
problem
those
permits
which
assumption, let us assume that each of them, we must have J3,B,.., + _¢ B,'.t +...+
according I have
we
are
no
other
already
have
several
_q B,-.q = o.
alge-
known.
I
singular
(8)
Only the constants B will not be the same for two different singular solutions. It is therefore not apparent that these two singular solutions must satisfy one and the same system of invariant relationships.
54
However,
this
is what
takes
place,
as we
shall
prove.
/52
In order to formulate our ideas, let us assumethat q = 4; the line of reasoning would be the samein the case of q > 4. Let us consider the n relationships ° Let
us
form
determinants be zero. If which clude
this
must only
Table by means
is not
the
be satisfied the x's and
the
If they are relationships
The
M's We
the
formed
are will
of
therefore
(i
,,_,
..
T of
the
4 n coefficients
of
the
four
case,
we
shall
columns
obtain
B.
in
one
are
the
equal we may
If all zero,
three 4
let
us
first
order
of
Table
or more
relationships
M, III+ M211t÷ M3H_+
M_H_=
T.
o.
(18)
of the assumes
coefficient of Sk to be identically
form that
(6"), which is all of the M's
of
the
form
minors the
of
minors
the of
first the
order
second
4 for
_f2 , 3 and
4 for
of Table
T are
identically
one. minors obtained by taking by cancelling the lines 1 and
143.
become 51'_ I11-I_I_Hz -I-_I_Hs = o.
This relationship the first member assumed
of
zero.
for M' I , 2 and
have
the must
have
Let M' I, M'2 , _f3 be three of these arbitrary columns in the table and
It will
of
table
to zero, let us consider three deduce the following from them
shall therefore have a relationship to our hypothesis, unless one only
identically
All
this
This relationship (18) must be identical, because the is one of the determinants of Table T, which I assume zero. We opposed
(17)
n).
by all the singular solutions, which will inwhich will not include the indeterminate B !s.
identically (17), and
minors
---
must be identical, because is one of the minors of the
to be
identically
(19) the coefficient first order of
of Sk in T which I
zero.
55
This would still only assumesthat all cally zero.
be a relationship of the form (6"), unless one of the minors of the second order _ are identi-
If this is the case, it will Bi, IIt-which
is
still
a relationship
becomeidentically
BaHt=o, of
the
form
(6").
It may therefore only be the case that all of the determinants of Table T vanish identically. We shall therefore have at least one relationship (and, consequently, a system of invariant relationships) which must be satisfied by all the singular solutions of equations (i).
tions
It may be immediately concluded (i) cannot be singular.
But this solutions. We
have
is not
Just
all;
defined
we may
the
expand
singular
integrals H i of equations (2) which and which correspond to q invariants of equations (I). In lar
the
solutions
same
way,
with
we
may
respect
equations the _''s.
with
(2) and
These integrals respect to the
of
our
of
the
a definite
I"12, ..., (2')
solutions
definition
solutions
q arbitrary
equations
all
with
are linear with (linear and of
provide
to I[I,
of by
that
of
of
equa-
singular
respect
to
/54
q
respect to the _'s the first order)
definition
of
the
singu-
integrals
Iiq obtained
must be homogeneous _'s and with respect
and of to the
by
replacing
the
the same order, _"s. They will
_'s
both be
whole polynomials with respect to these variables, but they will not be necessarily linear with respect to the _'s. They may therefore correspond to integral invariants of a higher order, or to integral invariants of the first order, but which are not linear.
not
In addition, these satisfy identically
I may then tionship (6) is solution.
56
integrals must a relationship
be of
different the form
-- i.e., (6), (6')
they must or (6").
state that a special solution S is singular if satisfied for the values of x which correspond
a relato this
We
shall
then
have lli:
The
quantity
Ak
of factors _, and the Bk. i s We
shall
is
a monomial
_2, are
..., _n' algebraic
first
set,
formed
no
shall
changes
Every fies
one
at
the
as
was
will one
The
and
role
products
all
periodic
this
would
In
is
me
reasoning
to
suitable
power,
pursued
above.
one
the
q
integrals
H i
invariant
relationships.
envisages
the
Bn+|.il_l+
We
satis-
integrals
in
the
as _,
to
only _i's
delve
be
may
products the
of
which
either the
reasons
singular
will
without
--
be
in
any will
the
form
for
solutions
/55
these
relationships.
repeated, Bk. i
to
solutions
invariant
--
into
respect
singular
quantities
the
be
with
these
algebraic
the or
B_+,.it$_+...+B2,,.it_n.
solutions, and
of
the
integrals
values
I
the
which
--
in
(i) to will
corresponding
afield
that the
(2)
include
still to
_'s be
that
and
be they
and
the
my
play
$i's,
t$ i.
my
belief
the
subject;
it
is
that
case
of
may
the
singular. equations
when
a We in
the
periodic
respect x's
to
are
all
system to
of
the
shall No.
integrals
this
that
according
of
with
return that
unlikely
satisfy --
q different
different
very
be
algebraic
shall
assume
problem
necessary
numbering
a non-singular
I
provisionally
three-body
would
order
and
from
shall
observing
of
notation
respect
far
only
solutions
equations with
too
meantime,
correct,
section
employ
q
of
the
preceding
the
if
need
of
cannot
take
relationships,
these
system
wish
invariant
If
of
respect
same,
proof
and
not
periodic
linear
number
_qBiq,
algebraic
singular
above
solutions
proposition
again
line
valid
the
same
_i
later.
the
a
certain
problem.
This
of
in
I do
three-body
of
the be
the
of
259.
this
of still
presented
the
to
a
above,
B2.t_l_...+Bn.i_a+
coefficients
same
to
Bt.i_l@
The
the
the
with
still
proof
changes.
done
in
system
definition
satisfy
of
form Iii=
integrals,
product
conclusion.
solution
are
the
_iBi.1+ _Bi.2+...+
made
same
same
results
following
The
be
the
singular and
These the
need
arrive
by
E_' $_' "''' _n raised functions of the x's.
Bi: and
Z B_,/A_.,
257.
which
are
the
x's,
replaced
by
solution.
57
By stating that these q integrals are constants, and by replacing the x's by the values corresponding to a periodic solution in the equations thus obtained, one will obtain q equations of form (5), but where the exponent _k will be zero. These q equations must therefore be included among equations (5). Therefore, in order that equations (i) include q different respect to the x's_ it ponents _k be zero for Let
us
now
try
to
integral invariants which are linear with is necessary that q of the characteristic exevery non-singular periodic solution. determine
7_ZAidx,+
the
integral
2B,_dz,_lz,=
These invariants will correspond to (2) which are quadratic with respect F($i)=
f
invariants
of
the
form
(_.
(7)
the integrals to the _'s.
of equations The integral
(i) and
const.
will correspond to the invariant (7); this integral must be quadratic with respect to the _'s and algebraic with respect to the x's. In this equation, let us replace the x's by the values corresponding to a nonsingular periodic solution. We shall have
V*(_,)-_ ¢o._t., where F* is a quadratic polynomial to the _'s, whose coefficients are
(8)
which is homogeneous periodic functions
It must be possible to deduce all equations (5) in the following manner.
equations
of
with of t.
the
form
respect
(8)
from
When dealing with a problem of dynamics -- in particular, in the case of the three-body problem -- we have seen that the characteristic exponents are pairwise equal and have the opposite sign. We can therefore group equations (5) by pairs. Let us set
When multiplying equations (5') and (5") by each other, we an equation of the form (8), and all equations of the form linear
combinations If we
therefore
of
the
assume
equations that
thus
equations
obtained. (i) have
of the equations of dynamics, and that they contain gate variables, we shall have p pairs of equations and (5"). Consequently, for each periodic solution,
58
will obtain (8) must be
the
canonical
form
p pairs of conjusimilar to (5') we shall have p
/56
equations
of
Let
us
the choose
combinations for all of the other nomial to the only
form
(8) which
one
are
equation
instance, periodic
linearly
from
these
defined
for
values
We
now
determine
a way that or even of problem,
the the
Let
us
of x which
attempting
now
try
i.e.,
those
having
where
F is
a linear
to the
equations
and
the
F* of
to
determine form
linear
function
at
the
the
be
present
invariants
a double
of
with respect which are
a periodic may
for poly-
made
/57
solution. in
such
algebraic functions of the x's, x's. I shall simply pose this it
the
of
selection
are the
solve
to
of
time.
the
second
order
--
integral
products
dxidx k
(the
of this linear function are naturally functions of the invariants of the second order will have the following
retain
their
of the second degree functions of the x's
correspond
whether
coefficients of known functions
without
p
F*($i). Let us follow the same procedure solutions. We shall then have a certain
F*($ i) which is homogeneous and _'s, whose coefficients will be
must
independent.
coefficients
x's). These significance.
Let us select equations (i) and (2) once again (we the numbering given in No. 257), and let us form
shall always in addition the
equations d?,.
They will may write
lead us to equations as follows
_,_ dXi
which
(2a)
_i.
are
similar
to
(5),
and
Ai.e_ t = z _)Oi_. only
order
According to the preceding chapter, the invariants will then correspond to those of the integrals of
and
which
algebraic Let
are
from
linear
with
equations
with
respect
(5) because
respect
to
to
the
x's.
F($_$_--
$k$_)
I
(5a)
They
(2a),
differ
which
the
the
letters
are
accented.
of the second (i), (2) and
determinants
59
be
one
of
these
integrals.
If
corresponding to a periodic having the form
the
F* will
and whose We the
In
now
(9), the
divided
a linear
coefficients
have
form
be one let us
be
into
like
us choose obtained.
periodic
solutions.
are only solution.
now
these coefficients of the x's. Let According
us now to No.
of equations nents become If,
6O
for
the
manner
of
respect
in which
return 29,
equations
(5a)
(5).
be
of
formed.
may
be
Let
B;,.e--=','= .X_,_0h,
(5a")
then
of
the form
the
and of
have
linear of
example,
these
of
all equations procedure for
of the form all other
a relationship
of the determinants. The be functions of the x's which
x's
corresponding
to a periodic
the
selection
be
or
two
exponents
made
so
that
the
known
functions
of
the
first
order.
(4),
or more
may
even
invariavts
equations
when
by (5"), (5a") by (5'), and having the form (9). Each
all other equations of the form those which thus may be formed.
functions
(5), is modified equal. nine
relationships
dynamics,
whether
to
all
t.
may
algebraic
the
of
determinants
(5a')
values
are
the
A;,.._,,' = z_,;0.,,
shall
determine
values
equation
solution,
one equation from among Let us follow the same
for
the
an
periodic
equations
We
to
functions
term will be a linear function of this linear function will
defined
We must
periodic
will give us one, linear combinations
Let (9) thus
by
obtain
(9)
pairs. Let us multiply (5a') We shall obtain an equation
pair of equations (9) will only be
whose first coefficients
with
to a given
of equations
pairs
of these subtract.
be
determined
case
replaced
will
6_6',) = const.,
function
will
relative
are we
/58 F'(_iLI.--
where
x's
solution,
and
consequently
characteristic
equal
zero,
we
that expo-
may
write
the corresponding equations (5) in the following
/59
form (i0)
P_.-= X,'_,O,h-, The quantity Pk designates constants for coefficients.
a whole
polynomial
with
respect
to
t, having
These polynomials are of the degree q - 1 at most. In order to specify this more precisely, the number of polynomials is q. The first may be reduced to a constant, the second is of degree one at most, the third is of degree two at most, and so on, and finally the last is of degree q - 1 at most. In
the
its maximum a derivative one, and so
case and of on.
in
which
In every case, the groups. In each group, stant, and each of them In
order
the
degree
of
this
last
polynomial
is equal to q - i, the polynomial before the last, the q - 2nd one the derivative
that
q polynomials may be the first polynomial is the derivative of
there
may
be
p linear
reaches
the last is of the q - ist
divided into several may be reduced to a conthe following.
integral
invariants,
it
is
not sufficient that p of the characteristic exponents are zero. It necessary that p of the polynomials Pk be reduced to constants (or, which is the same thing, that these polynomials be at least divided into
p
groups).
From cance
is
of In
the
point
equations No.
important.
216
(i0)
we
This
of view
of
where
defined
study,
Pk may
an
invariant
our
not
integral
has
the
what be
is
then
reduced
invariant
to
the
signifi-
a constant?
whose
role
with dx.
respect
is
very
form
t."+ tf[:,. where F and F I are functions x's, and linear with respect A having
similar the
invariant
following
which are algebraic to the differentials
corresponds
to
an
integral
to
of equations
the
(2)
form
where F and F I are functions x's, and linear with respect
which are algebraic to the _'s.
with
respect
to
the
61
pond
In to
this integral, if I replace the x's a periodic solution, we shall have F'.
where whose
the
values
now determined (ii) starting
Let us consider two constant, and the second derivative of the second. written
the
0i's
and
the
which
corres/60
const.,
(ii)
F* and F_ are functions which are linear with coefficients are periodic functions of t.
We have tionships of
where
tF_=
by
the manner in which with equations (i0).
we
respect
may
to the
obtain
all
_'s,
rela-
polynomials Pk, the first being reduced to a being of the first degree; the first is the The corresponding equations (I0) may be
,_,= z$,ot,
(I0')
A, + A, t= z_iO_,
(i0")
e'i's are
periodic
in
t.
We
may
thus
deduce
Z$iO_--tZ_,Oi = const., which
is
a relationship
of
the
form
(ii).
We should note that equation (i0'), raised to the square, provides us with a relationship of form (8), and that a relationship of form (9) may be deduced from equations (i0') and (I0"), that is, (E6_O.I)(E$}O'_)--(E$}Oi)(E$iO'_') 260. Let us apply this let us determine what may be of the several types studied That is:
tials
The first dx;
type:
linear
= const.
procedure to the three-body problem, and the maximum number of integral invariants, in the preceding section, for this problem.
invariants
with
respect
to
the
differen-
The second type: invariants where the function under the sign f is the square root of a second-degree polynomial with respect to the differentials of the x's; The with
62
third
respect
type:
invariants
to products
of
the
of
the
second
differentials
order, dxidXk;
which
are
linear
The fourth type: invariants having the form considered at the end of the preceding section -- i.e., having the form
These of
different
integrals
of
The
first
The
second
The
third
_i _ 'k _
_k _' i
The
types
equations type:
of
linear
type:
invariants
(2) and
integrals
quadratic linear
type:
correspond
(2a),
that with
integrals
with
with
different
to
_'s;
types
is: respect
integrals
to
/6__!
respect
respect
to
to the
the
_'s;
determinants
• '
fourth
integrals
type:
having
the
form
F -F-tF,, where
F and We
may
periodic
F 1 are
linear
assume
that
solutions
of
with it
the
respect
to
is extremely three-body
the
probable
problem
In the three-body problem, the number number of characteristic exponents is
the
_'s.
is
that
none
of
the
singular.
of degrees of freedom twelve. According to
is the
six;
ideas presented in No. 78, there are six, and six alone, which vanish; the six others are equal pairwise, and have the opposite sign. There are therefore six equations of form (I0) and six polynomials P., of which four are of degree zero and two are of degree one. Or, _ere are three pairs of equations tions having the form (i0'), Let each
type
us
therefore
there
will
having the form (5'), (5"), four equaand two equations having the form (i0").
determine
how
many
independent
I shall state more precisely what I mean. variants of the first type as independent
./'lq, or
n
invariants
of
the
second f_t,',:
or n
invariants
of
invariants
of
be.
the
third
.f,::
....
I do not
regard
n
in-
, .fF.,
type f,/t+'_:
...,
7f']_
type
63
or n invariants
of the fourth type fF,,
when there the form
is an
/62
fF=, ...,
identical
(Ft
relationship
-{:}+tF_),
between
'PlFI-_ 'P2F2-_"._.- -*-'PnFn
where
¢I,
¢2,
"'',
_n are
integrals
It is apparent that we cannot the first type, i.e., no more than already known.
FI,
F2,
...,
Fn having
o,
of equations have more the number
(11.
than four invariants of equations (I0')
of
We cannot have more than thirteen invarlants of the second type, of which three will come from the three pairs of equations having the form (5') and (5"I, and the six others will be obtained by means of the squares of the four equations (i0') and of their products by pairs. These last ten exist in actuality. However, they are not independent of the four invariants of the first type, since they may be deduced by the procedure variants.
which (5')
given
in No.
245.
We
may
therefore
have
three
We cannot have more than eleven Invariants of the three will come from the three pairs of equations (5"I. Six will be obtained by combining the four
by pairs; two will be with the corresponding
obtained equation
by combining (i0').
the
two
new
third type, of having the form equations (i0')
equations
(10"I
Seven of these invariants are known. One is the invariant No. 255; the six others are those which may be deduced from the equations (10'I, but they may not be regarded as independent of four invariants of the first type, since they may be deduced by procedure given in No. 247. We
type,
therefore
have
Finally, we may not i.e., no more than One
64
may
of
these
four
new
have more the number
Invariants
invariants
of
the
third
than two invariants of of equations (i0").
is known,
that
of No.
In-
256;
Jl of four the the
type.
the
we may
fourth
still
have a new invariant. It is probable that these new invariants, the possibility of which was not excluded in the preceding discussion, do not exist. However, /63 in order to prove this, we must resort to other procedures -- for example, procedures similar to those of the method advancedby Bruns. Use
in
261. The invarlant still another form. Let
us
set
of
of
Kepler
the
an arbitrary
fourth
system
a._, ,Iv dt Let curve
us
consider
the
type
of
in No.
canonical
256
may
be written
equations
4>',__ _:v.
-- dyi'
TFF -
following
(i)
d._t
integral
taken
along
an
arbitrary
the
curve
arc
J -- f(_, Let
Variables
us
assume
that
we
are
dy, + x_ _,
writing
+...
the
+ x,, dy,,).
equations
of
arc
along
which integration is performed, expressing the x's and the y's as a function of the parameter _, and that the values of this parameter which correspond to the ends of the arc are s 0 and _i. The integral will
J
equal
dx J d_.
Let us in the
assume that we are preceding chapter,
F 0 for
t = 0.
Then such
the x's,
as
dF F, dx'
dF dy'
We
shall
have
the ...,
y's
considering our curve which varies with time
and
will
., =.:L_d
be
_7/
the
functions
functions
dx-4-
of
of
arc and
the
_ and
llke the figure F may be reduced to
x's of
and
the
y's,
t.
f[XX dtdxjd¢_
or
65
when
integrating
by
parts
,1.)g'=o --o".... _0,
not
as
as
have
a factor. only
the
Therefore, terms
it in
else
_
cons[
in
the
periodic
just as in the the restricted problem,
the
we
have
following
in
the
fact that this third term is the only one which zero. If it is zero, the second term will also that it is the only one which includes t.
except
reduced variant,
which do terms in
_k
first does not include t, the second includes and the third includes it in the second power.
Due to the it must be to the fact
t 2, due
shows
fk or
first
includes be zero,
solution,
case of problem,
the threethe general
a quadratic
in-
form
.
than
f ¢_l_, and
the
this
of
term
If there invariant
in
_2
which
does
not
vanish
is a quadratic invariant, must vanish for all points
In other words, the meaning given
this periodic in No. 257.
corresponds other than that of the periodic
solution
must
be
to
this
invariant.
which is known, solution.
singular
in
the
sense
133
/131
v
There
would
be
an
exception,
were not independent of each between them. In this case, respect
to
the
n exponents
other, but if there were one the coefficient of t 2, which
ratic
form
could these
vanish without all of its coefficients being n variables will no longer be independent.
To addition
with
if
the
relationship is a quad-
n variables
zero,
since
sum up, in order that there may be other Ruadratic invariants_ in to those which we are acquainted with, it is necessary that all
periodic
solutions
It
is very
be
singular
unlikely
that
or particular. 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 _ be According
the quadratic form corresponding to to the preceding statements, _ may
A _f_, On the quantities
other
hand,
_
is
/_V?,
coefficients Following
are
_,t,_o,
o0-,
a quadratic _.c_,
whose
" "
are
the
the
variables
_2,
algebraic
_,,
one of include
form
;+,. with
respect
to
the
of x I , x2,
YI'
Y2"
_,
functions
x i and
these invariants. terms in
Yi which
we
shall
select.
/132 In
this
p_oblem, which I have called the restricted problem, two of the bodies describe concentric circumferences, and the third (whose mass is zero) moves in the plane of these circumferences. 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.
ordinates
of
the
Yl
to
designate
134
and
Y2
third
I shall body the
with
use
x I and
respect
projections
to of
x2
to
these
the
designate moving
absolute
the
axes, velocity
coand on
the
V
moving
axes. Let
us
then
set ,b :-- F + (o G,
where in
F
the
velocity The
and
G
designate
the
energy
motion,
and
where
absolute of
the
equations
two
take
first the
bodies
The
integral
Volume
I,
Under
this
_ = No.
is
page
that
which
the
the
four
assumption,
our
entails
four
could
be
Let we
shall
All
the
x
Sxi,
and
and
us
one
consider
function rotational
center
of
gravity.
o
than
"the
Jacobi
integral"
y ..--
_
will
a quadratic
form
in
_y:,
algebraic
are
be
in
related
by
x i the
and
Yi"
If
we
assume
One
of
them
relationship
cons[.,
condition
_
no
will
point
longer
become
of
the
f_=f_=
o.
a
be
independent.
ternary
periodic
quadratic
form. For
solution.
this
point,
have
the
expressions
_,_,
If
Z)',,
be
$y i will
eliminated,
common
else
expression
will
following
variables
et'b dxl
.....
nothing
Lr,.,
'I_
which
their
dj.j = dt
our
coefficients variables
area angular
23).
Z.r,, for
the the
form
ct,P -- J)'i '
eonst,
9,
and
designates
around
canonical
d.ri dt
(see
function _
we
set
6_
=
0,
(i)
will
therefore
_o_,
_,p_o
and
they
will
_, _ Therefore,
for
The
group
entire
a point
of
all
and of
terms
vanish
with
the
exception
of
_,1,2.
vanish
with
the
exception
of
_O'.
the
periodic
for
t 2 will
solution,
therefore
let
be
us
set
reduced,
for
135
/133
this
same
(see,
point,
supra,
to
Table
10')
and,
since
fl
=
f'l
= O,
may
be
reduced
to
Ct:_%'. The not even
terms
in
t 2 must
vanish.
The
latter
is
vanish for the point under consideration; when the condition _ = 0 is not imposed,
provide
tion,
terms
in
However, we have
the
only
one
which
all the others because _6@ and
does
are zero, 6_ 2 do not
t 2.
_a 0 is not
also
zero.
d:% d_o d/, - _-_ -
For
one
point
of
the
periodic
would
assume
that
having
the
solu-
d_o _ =o,
da0 but is a which
we
cannot
be
continuous does not
sure
may designate Consequently, in Keplerian
from
terms
which
in
it
However,
of
it
I
The
having
infinity occur.
Nevertheless, which body, i.e.,
of
may
-
de periodic
be
noted
only
This
solutions da0 _--
that
by _ as a factor, it may be noted motion. t 2 can
0.
vanish
i.e., that
if
includes
the
the mass 6a 0 vanishes
we
same
there period,
small
quantity
of for
the second _ = O,
H
may
have
follows
this
latter
equation
would
indicate
that
be
reduced
to a binary quadratic form and, consequently, that its discriminant is zero. Thus, the discriminant A of H must vanish for every point of every periodic solution. 288.
However,
an
algebraic
relationship
such
/134
as
_==-O
cannot be valid, unless every periodic solution. If
136
the
relationship
it is
reduced
--O
to
an identity,
for
every
point
of
is supplemented by two other relationships F = _, (where
B and
functions arbitrary
y
are
two
arbitrary
which were designated algebraic relationship
G= y
(3)
constants, in
the
and
preceding
F and
G
are
section)
the
and
two
a fourth
II= o, the number of solutions whatever the constants
Yi
(4)
of these four algebraic B and y may be.
equations
will
be
Let us now consider a periodic solution, and the variables will be developed in powers of _ in the following form Xl
--:
the
s_me
way,
"_
x i and
-F'.
l_x_+ .... . '-:y_0+ t_.s'_
Iy, In
Xl
limited
F will
be
(5)
developed
in powers
of
_,
and
we
shall
have F
and
G and
H will
be
independent
_
Fo-t-
of
tsPi
state
....
_.
The quantity A remains. It may is algebraic in x i and Yi under the algebraically on _. If we
-,-
be stated that this function, terms of the hypothesis, also
that
is an integral invariant, we will be led to certain include the coefficinets of H , their derivatives, of the differential equations of motion.
We
assumed
which depends
that
H is
an
algebraic
function
relationships which and the coefficients
of
the
xi's
and
the
Yi'S.
We may assume that this algebraic function is included as a special case /135 in a definite type, not containing _ explicitly, but depending algebraically on a certain number of arbitrary parameters. The quantity f /_-H will not be an integral invariant no matter what these when these parameters take on certain special When
stating
that
f _/_-is
an
integral
parameters may be, but values dependinE on _.
invariant,
one
is
led
only
to
certain algebraic equations between _ and these parameters. These equations must be compatible, and it is apparent that the parameters will be obtained as algebraic functions of _.
137
k.J
The coefficients
of the form _ and A will
also be algebraic in _.
The equation A = 0 is therefore algebraic in _, and we may assume that it has undergone a transformation in such a way that the first term is a whole polynomial in _. Wemay therefore write Ao +
A =
would
In addition, A 0 will not vanish, A would contain The
by to
the the
function
expansions fact that
A must
l_i-_-
_
A2 _ ....
be identically zero, a factor _ which,could
vanish
when
the
xi's
and
unless A is. be made to Yi'S
are
If A 0 vanish.
replaced
(5). It may then be developed in powers of _ and, due the term which is independent of _ must vanish we shall
have _0(x_,y_) -_o
We
should
now
point
out
that
we
(2')
must
have
(3')
i IL(
I may
.... also
call
E,
no matter
what
p
E < U_ 3 of
U(p). 0
151
The numbers _, 8, y, ...
are therefore all V E
smaller than
/150
Therefore, they cannot increase beyond any limit, and we may conclude that, starting with a certain order, all the terms are equal in the series of numbers _, 8, .... Let pe order.
us
Then and
u_P +I) so on.
and
the
UI(P+I) will
the
the
group
same
all
the
u(P X ) will
E l be
E is at
that
u_P)and
and
Let
assume
time.
terms
have
a P art
have
a part
i e consequent of
points
E l will
of
which
be
are
the
equal
in
to %,
common
in common
starting
which which
will will
with
be be
the
U 0(p+I) ' u_P +2) ,
E. are
group
part
of U 0, U_,
of points
which
U_,
...,
ad
are
part
of
inf.
TT
UX,
U_,
group
UX,
...,
of points
at
the
which
same
are
time.
part
It may
also
be
stated
that
E is
the
of (i)
U_"_", U_P*_'....
at the same time, since each of the regions U0, U_ is only a portion of the preceding region. In the same way, E l is the group of points which are part of
uip_,u_p_,, ..... at
the
same
time.
However, Each
term
Therefore
U_ p+I)
in E
series is
However, finite volume. xth
consequent
sam__.__e volume. coincide.
(2)
is (2)
a part
we
a part is
of
a part
El,
or
assumed that E Due to the fact
E l must E cannot
also
of
be
therefore
ux(P) , and of
the
u(P +2)
is
a part
corresponding
coincides
with
term
of in
- (p+l) uI series
E X.
is a certain region in space having a that the fluid is incompressible, its a certain he
region
a part
in space
of E X.
having
Therefore,
the E and
EX
If we assume that E is a certain region in space having a finite volume, it must be stated that E coincides with one of its consequents.
152
. (i).
_j
295. I shall
Following
confine
are
myself
some
theorems
to discussing U_,.
be
those
numbers
consequents _
are
U_,,
of U 0 which
arranged
which these
have
in order
of
are
all
but
theorems.
....
U_:,,
a part
and
... in
increasing
obvious,
Let
common
with
magnitude.
O 0. We
The
shall
have
V
Then
let Uy,,
be and
the
_
with
small
consequents U 0.
of
I have
as possible.
U0,
chosen We
shall
Uy,,
...,
which
have
these
numbers
Uy_
a part y
Let U
us employ the to designate
and
shall
U8__
in such
with
a way
each
that
y_
other is
as
V O.
notation given in No. 291 once the first consequent which has
with U0, U_ to designate sequent of U_ which has _, we
common
have , _ %jI+y_L
employ
in
this common part, U_ a part in common with
again, a part
and let us in common
to designate the U_. If 8 is not
first equal
conto
have
will
have
a part
in common
with
U 0.
Probabilities 296. We saw in No. 291 that infinite number of times. On the U 0 only these
a finite
latter
precisely,
number
molecules the
of must
probability
there other
times. be that
are molecules which cross U 0 an hand, there are others which cross
I plan
regarded one
to demonstrate
as
molecule
unusual crosses
or,
the to
fact
state
U 0 only
that
this
a finite
more number
of times is infinitely small, if it is assumed that this molecule is inside of U 0 at the initial instant of time. However, I must clarify the meaning which I am here attributing to the word probability. Let _(x, y, z) be a positive, arbitrary function of the three coordinates x, y, z. I may state that the probability that a molecule may be located within a certain volume at the time t = 0 is proportional to the integral J=:f_(x,y,z)dxdydz
153
extended over this volume. Consequently, it equals the integral divided by the sameintegral extended over the entire vessel V.
J
Wemay arbitrarily choose the function _, and the probability is thus absolutely definite. Since the trajectory of a molecule depends only on its initial position, the probability that a molecule behaves in a certain way is a completely definite quantity, as soon as the function ¢ has been chosen. Under this assumption, I shall simply set _ = i, and I shall try to find the probability p that a molecule does not cross the region U0 more than k times between the time -nT and the time zero. Therefore, let o0 be a region which is part of U0 and which is defined by the following property. Every molecule which will be within o0 at the initial instant of time will not cross U0 more than k times between the times -n'T and 0. If we assumethat our molecule is within desired probability will be
U0 at the time zero, the (1)
qO
P== Uo Let
be the region
n first consequents common to more than
of o 0. It is not Dossible k of the n + 1 regions O'O)
0"i)
,'_,) )
...)
therefore
a
O'n)
since, without this stipulation, every molecule this common region at the time zero would cross more than k times between the times -nT and 0. We
to have
which was located in o 0, and consequently U0,
have (_t-I-,)¢o < kV,
and,
consequently, P < (n.t-
kV ,)Uo"
No matter how small U 0 may be, or how large k may be, we may always take n large enough, so that the second term of this inequality is as /153 small as desired. Therefore, when n tends toward infinity, p tends toward zero.
154
Therefore the probability is infinitely small that a molecule which is located in the region U0 at the initial instant of time does not cross this region more than k times between the times --_ and 0. In the sameway, the probability is infinitely small that this molecule does not cross this region more than k times between the times 0 and +=. Let us now set n = k3 + x. The probability that our molecules does not cross U0 more than k times between the times -(k 3 + x)_ and 0, will be smaller than • _V (k3+x+I)U0 It tends toward zero when k increases indefinitely. The probability P that our molecule does not cross U0 an infinite number of times between the times --_ and 0 is therefore infinitely small. In reality, this probability P is the sum of the probabilities that the molecule crosses U0 only once, that it crosses U0 twice and only twice, that it crosses U0 three times and only three times, etc. However, the probability that the molecule crosses U0 k times and k times only, between the times --_ and O, is obviously smaller than the probability that _t will cross U0 k times or less than k times between the times -(k 3 + x)_ and 0 -- it is consequently smaller than kV (k_+z+ The
total
probability
P
V P < (x_-.2)Uo
terms
is
t)U0 therefore
_V + (x + 9)Uo +'"+
smaller kV (,_3___x +,)U,
The series of the second term is uniformly tends to zero when x tends to infinity.
series In
tends
to
zero,
and
P is
the
same
way,
the
probability
cule does and + _. The function
not
same _,
Equation
cross
results instead
infinitely
U 0 an infinite
are of
(i) must
obtained
setting then
be
_ =
is
+ ....
convergent. Therefore the
Each of the sum of the /154
small. infinitely
number
when
than
any
of
times
other
small between
choice
that the
is made
our
mole-
times
0
for
the
i.
replaced
by
the
following
155
k_2
J(_o) P -- J((u0)' where
J(o
o 0 and
U O,
and
J(U O)
designate
and
the
integral
J
extended
over
the
regions
respectively.
I shall assume that not become infinite, have
does then
we
O)
the function and I may
_ is assign
continuous; an upper
limit
consequently, _ to it.
it We
since
may
deduce
the
following p
_ + a_+_|.
However,
this
equality,
just
like
inequality
(2') (2)
itself,
may
be
satisfied
by arbitrarily large values of the xi's , because -- for very large values of the xi'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 preceding section are
late
168
conclude that not applicable.
In order to provide a better the integral invariant
the
considerations
determination
of
this,
given
let
in
us
the
calcu-
f
extending
it
over
a
dx', dz'_
region
@
@
defined h --¢
The
s's
of
equalities
of
(6).
Let
are
very
(5),
us
first
T
by
the
following
O) again, new encounter
if we extend the preceding will occur at
M 1.
quents
If
an arbitrary figure F 0 is of the different points of
called
the It
is
consequent evident
of
that
drawn in this half-plane, the conseF 0 will form a figure F 1 which will be
F O. Pl
and
z I are
continuous
functions
of
PO
and
z0 • Therefore,
the
consequent
curve, the consequent consequent of an area is connected n times. Let
us
now
of
a continuous
of a closed curve which is connected
assume
that
the
three
curve
will
will be a closed n times will be
functions
X,
Y
and
be
a continuous
curve, and the an area which
Z are
related
as
follows dMX a_
where
M
is
a positive,
Equations
(i)
then
uniform have
dMY dMZ *--dT--y+_=
function the
integral
o,
of x,
y,
z.
invariant
M.dxdyd_
177
and equations (2) have the following
invariant
/177
_I?d?dt,_dz.
Let
us
now
consider
the
dp
where
_
is
They
regarded obviously
as
equations
R
r[_
the
have
Z
d_o
independent the
(3)
variable.
integral
invariant (4)
(see
it
No.
is
3I t._o d? dt,J d:
253).
Since it was assumed a positive integral Let
F 0 be
an
above that invariant.
arbitrary
area (y
and
let
F1 be
Let
J0 be
its the
_o,
M,
_ and
located
in
p are
the
essentially
positive,
half-plane
x>o),
consequent. integral
(5) extended over the planar area F0, tended over the planar area F I .
and
let
Jl be
Then let _0 be the volume produced by the tated around the z axis by an infinitely small tegral (4) extended over _ will be J c. 0 0
it
is
tended
_I,
In the same way, let _I be the volume turned around the z axis by an angle over
_i will
be
The integral invariant and we must have
178
Thus, the consequent.
inteBral
same
integral
area F 0 when angle g, and
ex-
it is rothe in-
produced by the area F 1 when c, and the integral (4) ex-
Jl e. (4) must
Jo
its
the
=:
(5) has
have
the
same
value
for _0 as
for
Jl,
the
same
value
for
an
arbitrary
area
and
v
This is a new form of the basic property of integral
invariants.
306. Let us then assumea closed curve CQ located in the half-plane (y = 0, x > 0) and encompassingan area F0. Let CI be the consequent of CO• This will also be a closed curve which will encompassan area FI, and this area F1 will be the consequent of F0. If the integral (5), extended over F0 and over FI, has the value J0 and J1, we shall have from which it follows that F0 cannot be a part of FI, and Fl cannot be a part of F0. Four hypotheses may be formulated regarding the relative of the two closed curves CO and CI. i.
C1 is within
CO;
2.
COis within
CI;
3.
The two curves are outside of each other;
4.
The two curves intersect.
The equation J0 = Jl excludes the two first will
position
hypotheses.
If the third is also excluded, for whatever reason, the two curves definitely intersect.
For example, let us assumethat X, Y, Z depend on an arbitrary parameter _ and that for _ = 0, CO is its own consequent. For very small values of _, COwill differ very little from CI. Therefore, it could not happen that the two curves CO and C1 are outside of each other, and they must intersect. Invariant 307. invariant
Any curve curve.
Invariant point
of
the
which
curves half-plane,
may
will
be and
be
readily let
Curves its
own
consequent
formed.
Let
M 1 be
its
will
M 0 be
consequent.
an Let
be
called
an
arbitrary us
connect
M 0 to M 1 by an arc of an arbitrary curve C O . Let C1 be the consequent of C O , C 2 be the consequent of C I, and so on. The entire group of arcs of the curve CO, CI, C 2, ... will obviously constitute an invariant curve.
179
But we may also consider invariant more natural.
curves whose formation will
Let us assumethat equations (i) have a periodic solution. x = ?,(t), be
the
tions
equations _i are
I shall
curve
of
this
periodic assume
periodic
in that
y = ?2(0,
z = ?_(t)
solution,
in
t, having when
the
period
t increases
Equations (6) represent a curve. intersects the half-plane; this
be
Let
1179
(6)
such
a way
that
the
by
27.
func-
T.
by
T,
m
increases
Let M 0 be the point where point M 0 will obviously be
this its own
consequent.
close
Let us now assume that there are asymptotic to the periodic solution (6). Let x=
be
the
equations
of
these
The functions _i may efficients are themselves is a characteristic In equations
4,(t),
y = 42(t),
which
are
z = %(0
very
(7)
solutions. be developed in powers of Ae at, and the coperiodic functions of t. In this expression,
exponent,
(7),
solutions
the
and
three
A is
an
integration
coordinates
x,
y,
constant.
z are
therefore
ex-
pressed as a function of two parameters, A and t. These equations therefore 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 = O. The asymptotic certain curve which an invariant curve. 308. Y,
ceases
_.
Let
Z depend I shall to be
on
us the
assume closed
Let A 0 be For _ = O,
surface passes
consider
will intersect the half-plane through the point M 0 and which
along a is obviously
an
assume
parameter
invariant _,
as well
that for _ = 0, the for small values of
a point of K. the curve K is
curve as
K.
the
curve _.
I shall
curve K
that
X,
K.
is closed,
but
that
The position of this point will depend closed, so that, after having traversed
it
on
this curve starting with A0, one returns to the point A 0. If _ is very small, this will no longer be the case, but one will pass very close to A 0. Therefore, on the curve K there will be a curve arc which is
180
different from that where A0 is located, but which will pass very close to A0. Let B0 be the point of this curve arc which is closest to A0. /180 I shall
join
will
Let A 1 and be located
line
A0B 0 . We
must
AoMB 0 of A0B 0. In the
four
B 1 be on K.
the consequents of A 0 and B 0. These Let AIB 1 be the consequent curve of
consider
curve
What
A0B 0.
K,
will
order
to
points
the
closed
included its
consequent
define AI,
curve
between
our
A0,
C O which
A 0 and
B 0,
is and
two the
composed of
the
of
small
points small
the
arc
line
be?
ideas
more
B I, B 0 follow
precisely, each
other
let
us
assume
on K
in
the
that
order
AIAoBIB0.
curve
The consequent K and of the
C 1 of C O will be composed of small arc AIB I, the consequent
Several
hypotheses
i.
small
The
may
curvilinear
then
be
the of
arc the
AIMB 1 of the small line A0B 0.
formulated:
quadrilateral
AoBoAI_
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
A c A,
....
B,
Figure
I
This hypothesis must be rejected, because it integral J is larger in the case of Figure 1 for smaller in the case of Figure 2.
Figure
2
is apparent C 1 than for
that the CO, and
181
2. The arc AoAI or B0B1 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 corresponding to small values of the quantities which I have designated as Ae _t above.
On the other hand, the line A0B 0 does not have a double point, and the same must be true for its consequent AIB I. To sum up, we shall assume that the four sides of our quadrilateral do not have a double point. 3.
The
case
includes
will
then
arc
A0A 1 intersects
that
have
the
in which form
the
shown
the curve
in
4. have
The the
arc
A0B 0 intersects
form
shown
in
Figure
B0B I.
K would
Figure
Figure
then
arc
(As be
a special
closed.)
Our
case, curves
3.
3
its
consequent
AIB I.
Our
curves
will
4.
There are cases in which this hypothesis example, let us assume that X, Y, Z depend on for _ _ 0 the curve K is closed and that each consequent, so that for _ = 0 the four apexes coincide.
must be rejected. For one parameter _, and that of its points is its own of the quadrilateral
Then the four distances AoB0, AIBI, AIA0, BIB 0 will be infinitely small quantities if _ is the main infinitely small quantity. Let us assume that AIA 0 is an infinitely small quantity of the order p, A0B 0 an infinitely small quantity of the order q, and that q is
182
this
/IS2
V
A
A_
Figure
larger
than
4
p.
Since AIB I is the consequent of AoB0, the length of the arc AIB 1 must be of the order q. Then let C be one of the intersection points of A0B 0. In the mixtilinear triangle whose two sides are the lines AIA 0 and A0C , and part of AIBI, the others. must be The 5. AIA 0 and
It of
whose third side AIC is
should therefore the order q.
hypothesis
must
Two adjacent AIB I. It is
side is the larger than
be
of
the
therefore
be
order
sides of the quadrilateral then necessary that AoB0,
Figure
is
apparent
p,
that
A_
and
of the curve AIC which difference between the and
we
have
seen
that
is two it
rejected.
of AIBI, intersect K itself. If A_ is and A_ is the intersection of Al_with sequent of A_, and we shall obtain the
It
arc the
intersect_ for example, which is the antecedent
the intersection of A0B 0 with K, the arc AoAI, E l will be the confollowing figure.
5
A_ may
play
the
same
role
as
A 0 and
AI,
183
k_J
and
that
we
This
therefore
new
return
hypothesis
To sum up, the two hypotheses 2 and 4 must
to
must
the
therefore
case.
be
1183
rejected.
arcs A0A 1 and B0B I will intersect every be rejected, for one reason or another.
We must now examine the case follow one another in a different BoBIAoAI, AoAIBoB just studied.
first
I do not
differ
in which the order on K. essentially
points A1, The orders
from
that
time
that
A0, BI, B0 BIBoAIA 0,
which
we
have
Orders such as AIBIBoA 0, AIBoBIA0, AIBoAoBI, ... will not appear in the applications which follow. We shall always assume that, if _ is very small, the distances A0A I and B0B I are very small with respect to the
length The
no
longer
of order
the
arcs
AIAoBoBI,
discuss
AoMB 0 there 309. solution
will For
AoMB 0 or
them. be
or
the
It
is
a point
example,
AIMB I. equivalent apparent
which
let
us
assume
x = ?,(t),
and
asymptotic
_,
and
us
that
assume X,
be
if
its
that
y = ?2(t),
remain,
they
own
and
appear,
we
on
shall
the
arc
consequent.
equations
(i) have
a periodic
z = ._(t)
(6)
z = %(t).
(7)
solutions x = %(t),
Let
will
orders,
that
Y,
that Z may
equations be
y-=- _(t), (i) depend
developed
on
in powers
a very of
small
this
parameter
parameter.
For _ = 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 _t, with the coefficients themselves being periodic functions of p. However, the exponent
_ depends
on _;
let
Then for _ = 0 the functions and the solutions (7) may be
us
assume
_i will reduced
that
it vanishes
184
The 307.
_ = 0.
become periodic functions to periodic solutions.
The asymptotic surface intersects the half-plane curve C O which passes through the point M 0, which is the half-plane with the left curve (6).
No.
for
of
t,
along a certain the intersection
curve C O is obviously invariant, as I stated at Fo_ _ = 0, each of the points of C O is its own
the end of consequent.
of /184
In addition, = 0.
I shall assumethat the curve CO is closed for
Let us refer back to Chapter VII, Volume I. Wesaw from Nos. 107 on that, in the case of dynamics, the characteristic exponents may be developed in powers of _, and are equal pairwise and have the opposite sign. Weshall assumethat this is the case. In reality, we then have two asymptotic surfaces corresponding to the two equal exponents having opposite sign _ and -_. Wetherefore have two curves CO which will intersect at the point M0. Wemay distinguish
between four branches of the curve C_.
all
four
the
exponent
of
which a,
end C_
at
and
the C_
to
C" "_U_
C' all
C|
point
Mo;
the
exponent
C_
and
C'_ will
correspond
to
-_.
Bo
These
different
branches
Figure
6
of
curve
the
are
shown C "O is
branch C_ is the branch MoPoPIAoA I, the branch the branch C_ is the branch MoQIQ 0 and the branch MoRIRoBIB 0. These
four
branches
of
the
curve
are
in Figure the branch
C'_ is
obviously
the
6. The MoEoE I,
branch
invariant. I!
Now, for _ = 0, C_ is identical to C_, C_ is identical to C1, and (if we assume that the curve C O is closed for g " O, which we shall 0 call CO, ) these four branches of the curve will coincide on the closed 0 curve C0. It may
be
deduced
from
this
that,
for
very
small
_,
these
/185
branches
185
of the curve will differ very little from each other, that C'0 will deviate very little from C'I, C_ will deviate very little from CI," and that,if C_ is sufficiently extended, it will pass very close to C", if it is sufficiently extended. 1 I have indicated on the figure different points of these branches of the curve and their consequents. Thus, AI, BI, El, PI, QI, R1 are, respectively, the consequents of A0, B0, E0, P0, Q0, R0. Wewould first like to note that the points AI, A0, BI, B0 do follow each other (as we assumedat the beginning of No. 308) in the I
order AIAoBIB 0 when and Cy is traversed This the
invariant
closed In
As we
the invariant curve from A 1 to B 0.
curve
this
have
curve
is not
formed
closed,
but
of
it
the
two
differs
branches
very
CO
little
from
C_.
connection,
seen,
the
let
first
us
must
examine be
the
five
rejected.
hypotheses
The
second
of
will
No. no
308.
longer
occur.
It
could
only
occur
if
the
asymptotic
surface
(7) had
a double
line. We have stated that Therefore let us set
the
_i's
may
'I_g= ,l_t ° -;- Aeat,l,_
If satisfy
our surface had equations (i).
a double Actually,
an infinite number of lines that, if two layers of this could
we
only
be
one
of
these
where
A
three
identities
186
and
B are
two would
of Ae _t.
_-:k-'e_-_tq,_-_-....
lines.
a double
+_(t,
in powers
line, this double line would have to the asymptotic surface is produced by
time
t and
,I,,.= q,Jt, were
developed
satisfying these equations in such a way surface happen to intersect, the intersection
Since _i depends on the may show this by writing
If there identities
be
line,
A)=
have
A at
the
same
time,
have
to have
the
three
(i = ,, a, 3),
and where
to exist
parameter
A).
we would
4,_(t', B)
constants
the
t'
no matter
is
a function what
t may
of be.
t.
These
/is6
Performing
differentiation, d'Pt
we shall dq, i dr'
d--F=-yFWi However,
in
view
of
have
•
equations
d.t,, = x[+_(,, dt
(i),
we
shall
have
A), +.(t, A), +,(t, A)]
and in the same way d4:,t -- X[+,(C, dr'
from
which
it
follows
B),
+_(t',
B),
d'l_i
dr'
-_F = -dF' which
we
B)],
that
d¢i from
+_,(t',
-37 = "
have t'=t+h,
where
h We
is
a
would
constant. thus
,1,,°(t)
obtain
the following
+ A e _t,t,_ (t) + A' e TM q,', (t) = ,;,f(t
+
h)+
C e _t q,l (t + h) + ....
where C=
The
identity
must
be
valid
for
t =
Be _h.
--®,
from
which
it
follows
that
Ae =t= _e =t= op and
we
have
• ?(t)= ¢_(t + h),
from which we have h = 0 and
+_ (t)+ AeatOl (t)+-.., = _,_(t)+Ce=t,_l(t)+.. or
AOl (t)+X'*"+,'(O+...= C *f(t)_-Oe='¢_(t)+..
or,
setting
t =
--_,
we
have A = C = B.
187
J
Due
to
the
double
fact
that
the
two
values
A and
B
are
equal,
there
is no 1187
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 different
as we
the distances applications
Finally, the have seen.
fifth
AIA 0 and A0B 0. which follow.
hypothesis
Extension 310. tions
We
(i), but
formulated all
of
of
the
very
them
is
always
Preceding
special
are
not
This
is what
reduced
to
the
shall
first
do
in
one,
Results
hypotheses
equally
we
above
concerning
equa-
necessary.
Let us consider a region D which is simply connected and which is part of the half-plane (y = O, x > 0). Let us assume that we know arbitrarily that, if the point (x, y, z) is located at a point M 0 in this region at the initial instant of time, m will constantly increase from 0 to 2_ when t increases from 0 to to, in such a way that the curve satisfying equations (i) and passing through the point M 0 -is extended from this point M 0 up to its new intersection plane
--
is
Just
never
tangent
as in No.
M0, and it applicable
305,
to we
a plane may
is apparent that all to the figures which
then the are
passing define
through the
assuming that it with the half-
the
z axis.
consequent
of
preceding statements will located within the region
the
point
still D.
be
It will not be necessary that the curves satisfying equations (i) and intersecting the half-plane outside of D be subjected to the condition of never being tangent to a plane passing through the z axis. It will
no
longer
be
necessary
that
x = y = 0 be
a solution
of
equations
(i).
quent,
Then, the
The invariant
if C O is a closed curve inside of O and if CI is two curves will be outside of each other or will
results curves
given which
in No. do not
curve leaves the region will still be applicable this
D when it is sufficiently to the portion of this
applicable to the If even one invariant
extended, curve which
the results is within
region. 311.
188
308 will be equally leave the region D.
its conseintersect.
Let
us now
consider
a curved
surface
S which
is
simply
/188
v
connected, instead of a plane region D. Let us pass a curve Y satisfying equations (i) through a point M 0 of this curved surface, and let us extend this curve until it again intersects S. The new point of intersection M I may still be called the consequent of M 0.
other There
If we consider two points M 0 and M_ which are very close to each their consequents will be, in general, very close to each other. would be an exception if the point M i were located at the boundary
of S, or if the curve y touched the surface point M 0. Except for these exceptions, the tic functions of the coordinates of M 0.
at the point M I or at coordinates of M I are
the analy-
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 M 0 inside of D, intersects S at a point M I which is never located at the boundary of S -- so that the curve y does not touch S either at M 0 or at M I. Finally, I shall assume that this region D is simply connected. Let E, n and
us
adopt
_, for
a special
example,
system
and
of
coordinates
for which
I shall
which only
I shall
assume
the
i. When [_[ and IT][ are smaller than i, the rectangular x, y and z will be analytic and uniform functions of $, n and are periodic with the period 2_ with respect to _. 2. a point
No more than one system (x, y, z) in space, such
of values that
of
$,
n,
_ can
call following:
coordinates _, which
correspond
I_IT--
number.
By
-"
hypothesis,
the
exponent
al equals
7
pT
where
k
is
able
with
an
integer
number,
and
the
other
exponents
we
are
concerned
are
not
commensur-
= #227. , in general. T
The
difficulty
with
which
will
not
therefore
occur.
In
and
order
to avoid
this,
in No.
I assumed
that
(k integer
number)
(k integer
number)
not _t " :
pT
Special 335. assume
Let
only Let
246
330=
two
us
say
a few
degrees
us assume
that
of the
Cases
words
about
the
simplest
cases,
and
let
us
freedom. form
which
is
similar
to
that
which
I have
designated as U0, in the analysis of No. 331, is homogeneousof the third degree only in xI and x2. The
equation duo
dUi
d_', d::_ always
/246
has The
real
r_ots,
theorem
is Xl
as we
have
self-evident
duo
dU,
dx,
dx,
=o
(1)
seen. here,
since
this
or
three
real
to
clarify
equation
is of
the
--e
third
degree
assume
that If we
choosing _p2, the
will
in
x2
It
may
have
it has
only
one
in
order
w, =
a,?
cos?
-_ bt?
sin 9
x_=
a_?
cos?
+
sin?,
a
and
b
in
u0 :¢ U_
considered
then
the ratio
only
a maximum
This maximum and this sign, will correspond We
will
roots.
our
ideas.
that
U 1 is
Let
us
first
set
coefficients
have
one
then
and
b2?
such
a minimum
a way
in
No.
when
to
331
¢ varies
minimum, which are equal to values of _ which are
reduced
and far
from
0
to
2_.
which have opposite removed from _.
have Uo ! zU, = p_f(?)--zp,.
which
The function f(¢) has have opposite sign.
a maximum and a minimum which are The function U 0 + zU 1 then has:
For
z > 0,
a maximum
for
0 = 0 and
two
For
z
a minimum
for
p = 0
two maxima.
< 0,
and
Employing the English term, I shall use nate a point for which the first derivatives is neither a maximum or a minimum.
very
The same will hold small -- the terms Therefore,
have:
no
matter
true U0 + what
be,
the
and
minima.
the word minima to desigvanish, and where there
for the function V, zU 1 alone will have z may
equal
since -- if z is an influence.
differential
equations
will /247
247
v.)
z
0.
solution
0 and
unstable
Let
us
The
function
equal
now
pairwise In
assume f(_)
and
this
the
of
first
the
that
equation
will
have
have
case
of
U0 +
ZUl,
second
which
type,
(i) has
three
opposite
type,
and
and
consequently,
a maximum
for
p = 0,
and
six
For
z
a minimum
for
p = 0,
six
maxima.
be,
the
matter
what
z may
is stable
real
roots.
three
minima
for
which
are
V have:
z > 0,
No
stable;
signs.
For
< 0,
which
three
maxima
is
minima;
differential
equations
will
therefore
have : A
solution
Three below
of period
solutions
that,
from
of
T,
of
period
a certain
the pT,
point
of
first of
type_
the
which
second
view,
none
is
type. of
stable; We
these
shall
see
solutions
are
different.
let
Let us proceed us assume that In
this
case,
to a case which is U 0 is of the fourth equation
(I)
is
of
a little degree. the
fourth
always has at least two real roots according two or four. We then no longer have
but
varies
us
first
assume
The function from 0 to
this
maximum
First The
that
f(_) will 7, as well
A distinction
248
complicated,
degree,
to No.
331,
and,
and
since
it will
it
have
rather
Let
of
more
may and
case. functions
be
this
The
are
only
two
real
roots.
then have a maximum and a minimum as when _ varies from _ to 27.
drawn
between
two
cases,
depending
minimum.
maximum
U0 +
there
ZUl
and and
the
minimum
V have:
are
positive.
when
on
the
signs
For
z
>
For
z
< 0,
In the
0,
a maximum
a minimum
addition
to
differential
z > 0, stable
and and
case.
The
constants
For
z >
For
z
The
differential
0,
type,
Third
in
The
differential
For
z
For
z
> 0,
a
< 0,
a
for
to
examine
the
The
equations
then
have:
For
z
two
0,
type
type of
these
and
two
maxima.
which the
always
second
exists,
type
two
solutions,
the
minimum
for one
is
/248
is
negative.
minima;
two
an
unstable
the
first
solution type
of
which
the
is
negative.
have:
first
the
first
which
case
of
is
the
of
minima.
have
solution
then
of
now
must
=
itself
second
first
Of
always
of
solution
0,
p
equations
solution
the
positive,
=
the
maximum
and
V have:
p
for
minima
solutions
< 0.
is
and
the
We
z
equations
addition
The
of
zU 1
a minimuma
case.
solutions
of two
for
two
0.
have
a maximum
< 0,
P =
maximum
U 0 +
0,
solution
have any unstable.
The
second stable.
the
p =
for
equations
do not one is
Second
for
in
type
which
is
stable;
type
which
is
stable,
one
is
which
stable
equation
and
and
one
is
(i)
has
four
two
unstable.
real
roots.
of
the
which
are
For tions
of
second
upon
> 0,
second
a
solution
type
which
of are
the
first
unstable,
type and
which k
is
solutions
stable, of
the
h
solutions
second
type
stable;
z
< 0,
the
type
a
solution
second which
The
integer
the
signs
type are
the
h=k-=_;
the
first
are
type
stable,
which and
2
-
is
stable,
k
solutions
2
- h of
solu-
the
unstable.
numbers of
of which
h
maxima h=2,
and
k may
and A-=I; h--k=o;
the
take
the
minima h=_,
of k=o;
following
values,
depending
f(_): h==,,
k=o;
h=k-=l.
249
CHAPTER DIFFERENT
336.
FORMS
OF
THE
XXIX
PRINCIPLE
a double
these
ACTION
Let
y_, Y_, ..-, be
OF LEAST
series
variables.
of variables, Let
us
and
consider
J :
y,_ let
the
F be
an arbitrary
function
of
/249
integral
, t, ( -- F -t- Zyi dr;;,} -dr- / dr. o
The
variation
of
this
integral
may
( In
order
that
this
be
written
dri
variation
as
follows.
d g ri _ dt
may
vanish,
it
is necessary
that
we
have dx'i dt which provides not sufficient.
dF dyi '
us with the canonical If it is fulfilled,
ctyi dt
dF dxl
(i)
equations, we have
but
this
condition
is
::_[)_O.rt]lzto
and it is still necessary that the second term of this equation be zero. This is what occurs if we assume that the 6xi's are zero at the two limits -- i.e., that the initial and final values of the xi's are given. Under these conditions, action is minimum. Let ables,
us
and
perform let
us
the
the assume
integral
change that
J which
I have
in variables. they
have
Let
been
Ey_dx_-- Eyidxi=
chosen dS
designated
' xi, in
Yi' be such
the
as
the
new
a way
vari-
that (2)
is an exact differential. In this case, we have seen that the change 1250 in variables does not change the canonical form of the equations, and this result is an immediate consequence of different propositions which will be
presented
below.
Let J'= ft"(--F+Zy_)dt.
250
Wehave j,
where
S O and We
S 1 are
therefore
the
values
j =fas
dt = st-
of
the
So,
function
S for
t -
t O and
t = t 1.
have
_J'= _J + [_s]_:. If
the
canonical
equations
(i) are
(3)
satisfied,
we
have
t=t I _J = + [Ey_ _xl],=t.,
and,
consequently,
in view
of
(2) and _J'=
In the
the
same
way
relationship
that
(4')
is
(3),
[zy_
relationship equivalent
to
a-7 = a-_,' However, we have just seen that tions (i) are equivalent to equations knew -- that the change in variables of
the
(4)
'_t,
(4')
(4)
is
the
equations
at -
equivalent
to
equations
a_
(4) is equivalent to (4'). Equa(i'), which means -- as we already does not change the canonical form
equations.
The
action
final values the principle variables. The
J' will
be minimum
of the variables of least action
equations
(i)
entail
when
we
assume
that
the
initial
h
is
and
x i are given. Therefore, a new form of corresponds to each system of canonical
the
energy
integral
F = h where
(i),
(5)
a constant.
Up to the present, we have assumed that the two limits t O and t I are given. What would take place if these limits are regarded as /251 variables? Since F does not depend explicitly on time, we do not limit the conditions of generality by assuming that t o is constant, and by only increasing t I by _t I. For example, let us assume that t O = 0 and that,
after
values
at
the
the
variation, the variables, x i and Yi' have the same t time _ 1 (t I + 6t I) that they had at the time t before
the
variation.
251
Before the variation,
we shall have J :
--hh-bZ
ididt. dxl
fy
However, action
of
(T). q.e.d.
for
358• We must now relative _otion. The
irreversibility
determine
of
the
whether
equations
the
same
result
constitutes
is
still
a great
obtained
difference
289
kJ
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
longer have the basic property which I discussed section. However, there are other curves which which
have
preceding
this
property.
section
In
No.
340,
to we
This
is
remain
valid.
obtained
the
sufficient
following
curves
(i) will
in the I shall
for
for
the
the
no
preceding define, and
result
of
expression
the
of
/290
the
action J For
purposes
of simplification,
I shall
set
VH0
+ h
= F.
I shall
no
longer designate the coordinates by _ 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 m', but rather by
_,
removing
the
accent
which
has
J'=:[F from
which
we
become
useless.
then
have
v:d-x'--+ dr'-+-o_(xd: --y dx)],
have _J' _--f
[_v as + V ax _d.,"ds -_-,(r _'.r + _o(_z dy - _y dx)+
or,
I shall
integrating
by
_o(z Zdy
- y _d..)]
parts,
,
ds
+
The definitive expression of 5J' therefore includes two parts: a definite integral which must be taken between the same limits as the integral J', and a known part which I have placed between two brackets (according to common usage) with the indices 0 and i. This notation indicates that we must calculate the expression between the brackets for
the
two
the
second
Let
us
integration now term
tial equations which will be curves
limits,
assume
that
of
is set
(4)
and
the
must
expression
equal
to
which will be precisely satisfied by all of our
then
take
included
zero.
We
the
under
shall
the equations trajectories,
difference.
of motion, particularly
sign
f in
differenand the
(i).
These equations may be obtained in an infinite number because 8x and 6y are two entirely arbitrary functions.
290
the
obtain
of ways,
dF Wemay first assumethat _x = O, from which we have 6F =d_v _y" Dividing by _y ds, we may then write our equation as follows dF
cl_
dF
d.y
d t v
(6)
If,
on
the
contrary,
we
had
assumed
dF _
These two beforehand, dy ds
and
into
-_- 2to
dy ds
dz ds
dx d_s' respectively,
and
i r the
would
obtain
/291
d_x _- F -ds_f
readily be multiplied
following
determined them by
relationships
are
taken
account
arrive
at
an
-- \,is J
;_j'=
Let
AIBI,
the
curves
curves
A2B2,
Let
...,
action
end
r_,,r
i-d.r_Y
+(o(x_),_y_x
satisfy equation (4) becomes
) ]1 .
¢ls
o
a continuous
series
of
points
AIA2...An,
BIB2...B
two
these
which
arcs n
pertaining
form
two
to
continuous
C'.
AiBi, Ai+iBi+
by an infinitely Ai, x + 6x, and point
ds ,ts_
the curves (I), they will into account, relationship
[Fd,
AnB n be
(i), whose
C and
-ds 5_'s-_+
l;
identity.
If we therefore consider If we take this equation
(6).
1 be
of
small amount. let y + _y be
arcs
differ
from
each
Let x, y be the coordinates of the coordinates of the infinitely
other
the point adjacent
Ai+ 1 .
Let J' be relative If
a Is
the to
the
action relative the arc Ai+iBi+
angle
which
the
curve (I)] makes with the axis of satisfy the differential equation I" (cos
we
dF ds
_
_y = 0, we
equations are equivalent, as could If they are added after having
d_/
we
that
shall
a 3z
+
to I.
the
tangent the
sin _t _y)
+
arc
to
x's,
to (x
AiB i and
the and
_av -- y
curve if
8x ) =o,
the
J'
AiB I two
+
_J'
be
[which curves
the
is
a
C and
C'
(7)
have
291
v
and,
consequently, (A, B,) :_ (A, B,) .....
The curves defined by equation the orthogonal trajectories of section. We may therefore the curves shown
that
orthogonal There will
(A_ B,,).
(7) may therefore play the role which the curves (i) played in the preceding
consider figure by the dot-dash
(2) again, and we may assume line no longer represent these
trajectories, but rather the curves be nothing to change in the proof.
However, rectangular
one
point
triangle
is no
longer
clear.
defined
In
the
by
equation
infinitely
J292 (7).
small,
A3CB 3, I have (CBa)>(AaBa).
The the
triangle is no longer rectangular, definition of the action. Does It may
No. 341, equality
be
readily
seen
and we have seen therefore holds,
that
and in addition I have changed the inequality still exist?
this
equality
in No. 344 that they and our proof remains
equals
conditions
are fulfilled. valid.
(a) The
of
in-
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 cate_orM. 359. We must unstable solutions
make into
a few remarks regarding two categories.
the
classification
of
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 e is real, so that e at is real and positive, The a has
of
in -_
where
solutions as
of
be
divided
We may
the the
an imaginary
In the preceding the first class.
also
292
T is
set
into
period. second part,
class
are
those
that
e aT
is
so
for which
real
and
statements, we only considered Let us see whether those of the two
categories.
.
= _ =' + 'i"'
this
exponent
negative.
unstable solutions second class may
where _' is real.
where
_2,
changes We
_2,
_3
into
t +
then
have
Wemaythen set, just
and T.
as in No. 346"
_ = e_'e?_,
_, = e_'e_2 '
_ == e-_'e93 ,
_3 _- e-°ct+3,
_3 are functions These functions
of t which change will be real.
sign
when
/293
t
_,_ _ _, _ = _,_,G(t_.
change
The numerator and the sign when t changes
It quently
is therefore certain that the same holds
denominator into t + T. that these true for
of
G are
two
functions
functions
of
vanish,
t which
and
conse-
These last two functions satisfy the same linear differential equation of the second order, whose coefficients are periodic functions of t which have not become infinite. The coefficient of the second derivative may be reduced to a constant. These two functions cannot become zero at the same time, because if two integrals of tion become zero at the same time, they could only factor. However, _(t) is not a constant. The numerator and the denominator of _(t) zero, and do not become zero at the same time. sequently G(t)] may vanish and become infinite, All of the unstable second category. Apart preceding statements.
solutions from this,
the same linear equadiffer by a constant
therefore Therefore
both become _(t)[and con-
in question therefore belong to the there is nothing to be changed in the
293
CHAPTER XXX FORMULATION OF SOLUTIONS OF THESECOND TYPE 360. Weshall now demonstrate the manner in which periodic solutions
of
the
second
type
may
be
effectively
f294
formulated.
Let d:ri
dF
-)7 = ;if,'
be
a system
periodic
of
canonical
solution
of
first
We plan to study periodic from the solution of the The
analysis
may
if equations (i) in the variables. We
shall
increases
where
I
by
k I and
are
assume one
be
that
period,
k 2 are
at
least
then
assume
are
and
if
presented
this
were
in No.
208.
that
not
have
Y2
only will
for
two
purposes
by
which
of
a series
degrees
increase,
type
of
274,
361. the
or Let
canonical
to the us
the I =
true,
analysis
recall
changes
freedom.
respectively,
When
t
by
periodic O_
X!
_
solution
0 t
I could
if this variables
_1
_
were not the case, given in No. 202.
(2) may
be
the
presented results
to
01
perform
the
in No.
obtained
reduced
change
in variables
of periodic given /295
44.
in Nos.
273
to
277.
Let
equations d.v_ dF dt -- _i'
294
pro-
discussion,
of
Under this assumption, we shall see how the determination solutions of the second type is related either to the analysis in No.
a
numbers.
X
because,
they
second
form
I may first assume that k I = 0, because, could cause k I to vanish by the change in I may
that
(2)
a suitable
there
integer
assume
solutions of the first type (2).
to
Yl
us
(1)
y,- = +,(t).
simplified,
reduced
,t:_,,
type
x,-= ?,.(t),
ceed
dF
at Let
equations.
the
dv¢_
dyt _
--
dF dx_
(i-,,
_, ...;_)
(1)
include a parameter X, and let us assumethat they have one periodic solution c/= of period
TO,
corresponding
?i(t), to
yi = q_/(t),
the
corresponding to X = 0. Equations series having the following form; of the quantities ._,
Ake=_t,
value
(2)
C O of
(i) will be these series
h',e-=_t
(k -= ,, 2 ....
the
energy
constant
and
formally satisfied by will develop in powers
, n--
,).
The coefficients will be periodic functions of t + h, depending on the energy constant C. The period T will also depend on C and the products AkA _. It may be reduced to T O for C := Co,
), =o.
The exponents _k are constants which may be developed X and the products AkA_, and in addition they depend on reduced to the characteristic exponents of the solution
of be
The
Ak'S , the
real,
of ak
AkA_- = o,
When and
studying we made
In order the second are purely
remove
and
Ax.A2 = o,
h's
integration
asymptotic one of the
the
order
are
that
exponent
we _ be
may
two has
)_ = o. constants.
solutions, we assumed two constants A vanish.
to apply the same results type, we shall assume, on imaginary.
I shall assume only the index k which In
that
A_'s
C = Co,
degrees become
obtain
in powers C. They may (2) for
that
the ak'S
to a study of periodic the contrary, that the
of freedom, useless.
which
allows
were
solutions exponents
me
to
/296
periodic
commensurable
solutions, it is necessary 27 with-_. If our series were con-
vergent, this condition would be sufficient. However, they are divergent, and only satisfy equations (2) from the formal point of view. A more detailed discussion is therefore necessary. A method similar to that employed in Nos. 211 and 218 could be applied. We would thus obtain series which would have the and 277 as the series of M. 127. By solutions different
same relationship Bohlin have with
with those
those given
given in Nos. 273 in Nos. 125 and
an indirect method, we would thus fall back on the of the second type. However, I prefer to proceed manner.
periodic in a
295
kj
Effective
Formulation
of
the
Solutions
362. By performing the changes in variables presented in Nos. 209, 210, 273, 274, which are always applicable when we have a system of canonical equations having a periodic solution, we may change our equations to the form of the equations presented in No. 274. In this section, we have formulated the following equations (page 95)
.... dt
_z',
dx, v=,
-dF ....
(3)
F' = F; + _v; + ,, F; + .... where
F' p is
a whole
polynomial
of
degree p + 2 if it is assumed ' is of the second order. and x 2
periodic
functions
Just become
as
whose
in Section
useless We
of y_
then
and
No.
shall
assume
!
x2,
which
that x_ and y_ The coefficients period
274,
write
the
!
in x I, y_
we
F,
following
is
remove
xi,
(see
be
homogeneous
of the first order, this polynomial are
2_.
shall
Fp,
are of
will
Yi
pages
the
accents
have
F' , F; , xi, '
instead
of
q7,
99)
98,
which
Yi" '
Fo=Ilx2+2B_tY,, where
H
and
B
are
constants.
I could
also
set
H
= i, but
I shall
not
do
this. Just
The
as
on page
equations
will
99,
let
retain
us
the
other
respect
which
terms to i
Our we
v
u is
This
296
...,
will
respect
and we
shall
have
2Bu;
be
periodic
of period
2_,
both
with
to Y2.
where the parameter e plays we may employ the procedure
However, and 44.
F2,
with
IIxt+
form,
equations will then have the form which is similar to that have studied several times, and in particular, in Nos. 13,
125, etc., Therefore, tions.
x2 No.
FI,
and
set
canonical Fo=
The
then
there zero,
fact
is
one
which
compels
me
obstacle.
is
to
precisely
assume
the role of the parameter given in No. 44 for these
The
Hessian
one
of
that
the
F depends
of F 0 with exceptions
on
respect given
a certain
42,
_. equa-
to
in
parameter
%, and we shall carry out the development at the same time in powers of X and of e. Wesaw in Chapter XXVlII that when studying periodic solutions of the second type it is always convenient to introduce a similar parameter, because the property of being reduced to a solution of the first type for X = 0, and of differing from it for X _ 0, characterizes solutions of the second type. To facilitate the discussion, instead of an arbitrary parameter, I shall introduce two parameters, which I shall call X and _. Weshall therefore assumethat the different coefficients of F may be developed in powers of two parameters % and p, and that for _ = %= 0, H and 2B may real number.
of
c,
I shall in the
where %1, X2, termined, but which follow.
No.
be
reduced
to -i
assume that X and following form
..., are I reserve
and
to -in,
_ may
be
where
developed
constants which I shall the right to determine
Under this assumption, let us 44 step by step. We shall set
n
follow
the
is
a commensurable,
in
increasing
provisionally them in the
computation
powers
leave undecomputations
presented
in
/298
x, = _o -+c-t, + _'_,÷ .... u = Uo -_-"u, + _' u,+ ....
These
of
t;
where below.
formulas
are
similar
The _k'S, nk'S , Uk'S , and $0 and u 0 are constants,
_ is
an
integration
to the
formulas
the Vk'S are and we have
constant
(4)
which
(2)
therefore
I shall
of No.
44.
periodic
determine
more
functions
completely
Instead of %, p, x2, Y2, u and _ let us substitute their expansions in powers of E in F. Then, F may be equally developed in powers of E, and we shall have
if we
I would
like
to point
assume
that
_p and
Up
out are
that of
_k
is homogeneous
degree
p +
2,
_p
of and
degree Vp
are
k + of
2,
the
297
degree
p, It
Xp
is
and
_p
is
therefore
of
degree
a whole
p.
polynomial
with
respect
to
(p >o), and
with
respect
These
last
Finally,
the
q0 whose
period
In
to
two
quantities
coefficients is
addition,
may
of
be
this
assumed
to be
polynomial
are
on
the
order
periodic
of
functions
i. of
2_. we
shall
obtain
,l,m= O_--- Sk-- f*tuk-i- ),k Ilo_o+ "2Bog*kuo, where
H0
assume only
and
B0
that
we
are
the
values
dH dB -= 0 d> dX
have
dH _-_
of for
dB _
and X = _
for
X = _
= 0.)
In
= 0.
addition,
Ok
depends
on _p, %, Our
differential
For
k = 0,
they
up, %,
equations
d_.k d'l'l¢ _-l-[= d_-_' may
be
demonstrate
the
d,l',, -d;_7'
duo
_ is
fact
a constant
be
which
must
v_
be
:
dt -- in. u 0 are
inl
+
constants,
and
us
develop
xl
and
Yl
determined.
in powers
of
+q', + i
The pansions
expansions
(4')
y,
may
:=
P 2 t 4-e_' t _- _ _'-_ i t r,o
be
that
by,
We may advantageously add other equations having equations (4) and (5), which are only transformations Let
(5)
d'l'k du-_"
dvo
=t;
_0 and t,
dr j, _ dt ----
to
_
that
/299
written
du_ d,l,_ --0{ = -d('o'
d_jo
=o;
"_o -=
where
(pSk--J).
then
reduced
dt -- dt They
Z_, ILp
may
dr,_. _ dt
cl_u
directly
E, and
a similar of them.
form
+..-, '
•
then
find
that
Ck is
a whole
to
let
(4')
.-
concluded
from
the
two
last
(4). We
298
(We may
polynomial
with
respect
to the
ex-
quantities _p, %, _, _, lp, _p (writing and that that
this
polynomial
is homogeneous
of
degree
Sp is of degree _,_ is of degree %, Ip,ep is of degree We
then
have
the
following
which
are
We
form
as
equivalent
may
note
to
d+,
dq,_ d-_f'
that
_k with
respect
-- d_' o '
the
last
two
the
a_4_
d+_
dt
a_o '
are
(6).
having
Using
the
the
same
same
convention
find that they are homogeneous, of order k, and the last two of
have
us
replace
replace
nO
that
we
have
set
We
thus
have
by
t,
_0,
in
_0 by
equations
k = i,
the
and
six
d'_,
(5)
d;]'
1
Let us first a homogeneous,
dot
and
+=L
and
(5'),
at
the
in which
them
to
same
it must
determine
time be _i,
let
us
assumed nl,
Ul,
equations
dO,
d$1
dr, dO1 -dr = -- duo
2 _1 80,
du, in dr_--_o +_B°llt _[tZl
values,
us employ
following _lq_ 1
d_', dO, dt -- dr/o
these
let
dui dO, -di- = -_o'
is
assume
(5).
_ _o= _oe-¢nit Let
we
(5')
polynomials
quantities
employed above regarding the degrees, we the first of the order k + 2, the second the order k + i. We
2, if
p+% p+,, p.
equations
d,_k d,I,# d,_ d_"-;'-'d--_d_0
to
k +
(6)
equations
a_. d[
nO separately),
duo dO, dr_'o -- d7l_ duo
consider the second whole polynomial of
dO,
. ,, tn_t+_Bo
(7) _,_o,
dO_
of these equations. The second term the third degree with respect to
299
/300
01
whose
coefficients Since
n
is
are
periodic
The
period
multiples
of
Our the
will
2n
second
following
functions
commensurable,
function of t on which equals t, and by means
as
our
it depends of £'0 and
be
are
can
of
second
nO
term
in two ways: n_ which are
a multiple
there
term
"_0,
will
period
also
he
by means functions
of
2_,
i.e.,
in
the
denominator
be
developed
units
therefore
= % of
a periodic
of no which of nt + _.
it will
in
2_.
equal
as many
of n. Fourier
series
in
form
z ,'__/'J'___',_'+=I', where p and lute value,
This
q are since
integer numbers. our second term
is
(8)
However, q does not exceed a polynomial of the third
3 in absodegree.
As a result, in general the mean value of the second term is mean value will be obtained by retaining the terms which are
pendent
of
t in
the
series
(8),
zero. inde-
i.e., p+qn---o.
due
I have stated that JqJ can not exceed 3. I would like to to the fact that our second term is a whole and homogeneous
of degree 3 in _0, _0 and second term cannot contain
n_, it _'0 and
q must
take
be
odd
and
Therefore,
can
we
only
can
only
is @0
assumed that except in an
one
of
1 or
3.
the
values
add that, polynomial
_0 is of degree 2 or a uneven power -- i.e., +_i or !3.
have p_qn_o
if
the
denominator
We number,
the
i. fact
shall but
of n
exclude
two
equals
the
cases
first
remain
hypothesis
for
our
which
would
make
n a whole
consideration:
The denominator of n does not equal 3. that the mean value of the second term
In this is zero,
case, due to the equation
will immediately provide us with _I by simple quadrature. Then $i is determined up to a constant which I shall call YI, and this constant remains undetermined up to a new order. It should be pointed out that the
same 2.
be
30O
holds The
integrated,
true
for_.
denominator it
of n
is necessary
equals that
3. the
In
order
value
of
that the
the
second
equation term
be
may zero.
For this purpose, we shall employ the constant _. have
Let [ @I] be the meanvalue of @ 2. _
r__(_l
,z[o,l
nLd.,,o.l= and
we
shall
therefore
determine
Weshould point out that we
d_
_
by
;
the
equation
d[O,l (9) and
a
quadrature
Let may
be
us now
pursued
polynomial n
is not
d@l
is
of an
will
take for
the
then
the
this
first
us with
equation
equation.
third
integer
provide
degree,
number,
(7).
However, but
we
rather
shall
be
_I,
up
The
of
a constant
same dO 1
since
_0
the
certain
to
line is no
first
that
of
reasoning
longer
degree,
the
YI.
mean
and value
a since of
/302
zero.
d_0 It is therefore sufficient for us to take second term may have a mean value of zero, and determined to a constant 61 . Let us now pass written as follows
on
to
-
the
last
a-7
_
d'_ 'j
that
The second integration
firstequation term of the
They
may
be
+ =Bq_,_;,
does not include a term second equation does not
In order that the
containing e int, include a term
e -int .
dut
be discussed more readily by considering (7) which are equivalent to the last two,
dot,
dr,
at = -_ "
With
(7).
the be
dO,
This double condition could these third and fourth equations and which may be written
It
equations
terms are the known periodic functions of t. may be possible, it is therefore sufficient
second term of the and that the second containing
two
%1 = 0 in order that in order that nl may
is necessary respect
to
that the
the
mean
first
of
-_
dot
= - _ values
these
-- _ _, Bo. of
the
equations,
second the
terms
be
condition
zero. is
301
k.J
fulfilled
by
itself,
and
we
have
[e0 ] a[0,] dr,| = This latter expression denominator of n equals 3, identically The
is and
d---_"
zero because of in the opposite
zero. second
condition
may
be
written
e[o,] = duo
If
the
denominator
of n
On
the
other
if
vide
equation (9), if the case because [01] is
us with
_I
hand,
--_tBo-
equals
the
= 0, because
3,
it will
denominator
[01]
is
provide
does
not
identically
us with
equal
3,
_l" it will
pro-
zero. /303
and
Thus, of _.
we may see that _I, nl, _'I, @I They may therefore be developed
following
are periodic in Fourier
functions of series in the
t
form A _iIpt+qnt+q_)
However, having
the
we
may
following
d_
shall
a few words
more.
We
must
deal
with
equations
form
_
dt and we
add
derive
_r Aei(pt+qnt+q_)
the
d_ d-_ + ittN = Y = Y,Beapt+q.t+q_!,
1
following A t(lat+qnt÷qml
=E_(p+qn)
E _(p+qn+n)
_= where
y and
¥'
are
Therefore, respect
integration
if X and
Y
zero,
will
y and _ and Let
computed.
302
+q_)
.Jr_ ._' _-Int
whole
and
homogeneous
polynomials
with
to
same
stants
+qnt
_s
constants.
are
_/_o, the
el(pt
q-
us
hold y'
n will apply Due
true
are
for
zero. still
these
to the
_o eilnt+=_, g and If
be
whole
principles fact
n,
it is
that
¢ ge-t_nt+=' unless
not
it
assumed
polynomials, to
the
is
assumed
that but
quantities
these
not
that
the
constants
conare
homogeneous.
which
we
have
just
are polynomials ployed regarding
which, degrees,
dO1
dOi
dOj
d_t
a_o'
_S,,0' 7if'
a_;
according are
to the
of I,
the
same
will
hold
true
the convention which following degrees,
3,
a,
we have respectively
em-
2,
for
_,, _,, _',,_',. When we substitute the values of these quantities which are, respectively, of degrees i, 3, 2, 2, instead of these quantities in 92, it may be seen that @2 becomes a polynomial of the _urth degree, and that dO,
will
be
polynomials
of
the
dO_
following
dO,
/3O4
dOl
degrees,
respectively
2, 5, 3, 3. We
may
therefore
formulate
Equations (5) and (5') q_ from place to place.
g_, value zero.
of
Let
will
be
the
us
second
assume
term
that
polynomials
of
one
this
respect
is
Let us assume less than k.
know
this
result.
not
k,
equations
occur.
(5) were
It may
be
gk' qk' the mean
different
stated
from
that
degrees
k+l,
k+l
to
the coefficients of t of period
We
the
following
_' where tions
of
does
k+_,
with
of
enable us to compute the unknowns This would only be prevented if
of
the
a generalization
that
of 2_.
that
@k
is
_ei{nt+_', V/_o¢-i(nt÷_',
these
this
is
a whole _q,
polynomials
valid
for
polynomial _q,
_,
_
are
every
of (q 0 any for % < 0. In the second case, the opposite will
equation
taining
s2.
to
determine
which We
which
relates
shall
_
of
these
two
cases
to u 0, restricting
is valid,
ourselves
to
let terms
_ IdOl]. 2B0 [auo]'
_, [dO_']. X ----- Hoo[ d_oJ
rd02] z mayfirst observe thatt I andtd 4, terms having the following form ei_pt
,qtzt
* q_l
which may be included in the second can only be independent of t if
term
since
qn must
lq[ cannot
exceed
4 and
since
Thus, the second terms of equations functions of _0 and u 0. The coefficients absolute constants which are independent
However, u 0 must be positive; equations (21) added to inequality
in
one
be
of
the
equations
an integer
(21)
number.
(21) are linear and homogeneous of these linear functions are of _.
otherwise V_would u 0 > 0 will determine
be imaginary. the sign of
I need only point out that this sign does not depend on _, tions (21) do not depend on it. We have seen that the equation termines _ has two solutions which are actually different
The %.
since equawhich de-
7_
In
conformance
which will be of them. The
It of
_2_ 0 and E V_.
E V_
of
generality,
our
formulas
Out positive.
of
are
we
the
may
assume
any
way
our
two
systems
real,
a periodic
by
the
will
both
relationships
however.
Without
that V_o changing of
solution
% is suitably chosen corresponds does not depend on _, and these
% > 0 and
from
case,
in
we
statements,
be
imaginary
that two periodic solutions correspond for _, since two systems of values for
obtained
is not
Therefore,
Two
preceding
both be real for will hold true.
first appears the equation
This
the
real if the sign of choice of this sign
solutions will or the opposite
tion
with
between
values,
into there
_,
is
_,
the
because
we
and
_ into
only
one
for
type
for
% > 0
-_,
for
%
< 0,
to each soluthe unknowns /321
restricting
is positive, V_
to each two
e2_Q
and--
conditions
do
not
change
_ + 7.
which
_
is
have:
periodic
solutions
of
the
second
(or
for
_ 0).
319
k.2
Let us again employ the notation given in Chapter XXVIII and, in particular, that given in No. 331. U1 may be reduced to p2, and corresponds to the term containing xlY 1 which appears in 0Q. U0 may be reduced to a constant factor multiplied
by p4, corresponding
[d2 and l
[d_q
to
terms
The
first
following
form
LT oJ
fromL_u0j
coming
term
of W which
may
not
be
reduced
to
a power
U1 has
the
which
must
p,_ _=[A. cos(k -.- _)? ÷ B]
and
comes
from
Ok+ 2.
The
function
the
role
play
whose
of
the
maxima
and
minima
we
on page
247,
study,
and
function Uo__ =U 1 = p,/(?
studied
must
will
have
the
zp'
following
form
Apk+' cos(k _-:_)?+ P P_--zp', where
P We
is a whole have
n equals
2,
polynomial
disregarded 3 or
in p2 with
the
particular
they
Let
Then
[O2],
will
include
The
which only
us
assume
that
of this
F_ %l' F_ 0;
of-_,
to may
zero
the
fact
that
occur: solution
for
x 0;
one
solution
for
% Q; two solutions
for
The function U0 + zU1 given on page 247 becomes p_(Acos4?÷ B)--_p,. Let us now assumethat the denominator of n equals 3. The expansion of _ in powers of e then begins with a term containing E_0, so that if we set _ = %, we shall obtain c2_0 and e_/_- in series which may be developed in powers of %, and no longer of V_. The
sign
of
negative
V_will for
will
be
mainly
Therefore, positive,
The
function
on
_,
if
if
of the % < 0.
U0 +
the
it
is positive
for_=
D 0 it
= WO + _.
second
zU I given
type
on
denominator
for
page
Ap_cos3$-Finally,
and
if it is always convenient for we shall readily find that we
A real solution second type for
the
_
depend
of n
us to have:
assume
% > 0 and
a real
that V_
solution
is
of
247 becomes zpL
equals
2,
[02]
_|d02|
Fd02]
include
'Ld_0j ' Ldu0J' terms
containing The
equation
e -+4i_, e +2i_. for _
/323
takes
the
AcosC4w÷ and
it has
eight
form
B)_-A'cos(=_ ÷ B')=o
solutions 3_ 2
2
m_, _,÷_,
_-_-_,
--, m_-t- 3_ 2
Of
the
two
terms
-_0 and
_I,
at
least
one
is
real.
The following hypotheses are (0, 4), (2, 0), (1, l), (0, 2).
possible:
(4_
The periodic
parenthesis the second
represents is the same
first number between solutions for % > 0,
the and
0),
(3,
i),
(2,
2),
(i,
3),
the number of number for I < 0.
321
The function given on page 247 becomes A_cos_c?-'-B?_cos2_-+-,
Application 368.
Let
us
return
to
C_s;n2?+D?
to Equations the
dxl
canonical dF
Just periodic meter _
as in
No.
13,
No.
function of the in the following
42,
y's, form
t"
of No.
13
equations
dyt _
,_-7--_z,'
_-z_
of
dynamics:
dF
(1)
-a_----u_"
No.
125,
which
may
etc., be
I shall
developed
assume
that
F is
of
a para-
in powers
a
F=Fo+MFI+... and
that We
tions
F 0 depends saw
of
only
in No.
the
42
first
on
that
the
x's.
these
equations
zi = ?i(t),
where
the Let
There and
functions us
have We
the two
opposite saw
_i
consider
Let T be will be
in
and
one
an
infinite
number
of
solu-
_i may
of
yt = +i(t)
be
these
(2)
developed
solutions
in
signs,
where IV
that
When _ varies let us assume
we
may
assume
a depends
on
increasing
powers
of
_. /324
(2).
period, and let _ be one of of them, which are different
Chapter
powers of _/_. _. For _ = _0, to 2ni_.
have
type
the characteristic from zero, which two
_,
degrees
and
may
continuously, the same that aT is commensurable
of be
exponents. are equal
freedom.
developed
will hold true with 2i_ and
in for equal
We may conclude from this that, for _ which is close to _0, there are solutions of the second type, which are derived from (2) and whose period is
(k +
seen sign,
2)T,
where
k +
2 designates
the
denominator
If we put aside the cases in which k + that two of these solutions exist when and
that
they
do not
exist
when
I
of n.
2 equals 2, 3, or 4, we have I (here _ - _0) has a certain
(here
_ - _0)
has
the
opposite
sign. I have regarded,
322
stated
and
I may
that do
the this
cases without
in which causing
k + any
2 = 2,
3, 4 have
inconvenience.
been The
dis-
following aT _i_
may be values
developed in powers of _, n is therefore
larger
than We
itself
therefore the
occur
for
Which
of
have
two
solutions _
of
the
second
in
order
and
these
to
the
two
hypotheses
the
canonical
_
> _0,
or
this
Q, which
depends
LduoJ
sign,
we
considerations take
term
shall
will
a simple
case,
F = x2 _xT+
_cosyt
not
need
to
formulate
this
suffice. which
will
be
that
presented
/325
dyi dt
dF dx_
_
yields
The
function
dY_ --d-'['=--',
o,
S of
Jacobi
dz, d---_-=
may
S = z°y, two
constants
x02 and
l
where
for
equations
dz, d--7--=
with
only
is valid?
da"i dF dt -- dyi ' which
occur
the sign of a certain of u 0 and _0 in
determine
following
369. Let us first No. 199. Let us set
with
type
< _0"
Everything depends on on the coefficients
In
small definitely
hypotheses:
d_oJ'
term,
For is
4.
Either they
of _/_, and vanishes with _ very small, and its denominator
A and
yO
are
C.
x,=¢C-_cosy,,
two
new
be
x_ = x °,
integration
dyl --_--------2x,.
(i)
written
÷f_C
We may
. _ s,ny,,
-- _ cosy, derive
dr,
the
/
following
dy,
y, = -- t+y °, A--t= 2(C--_cosyi
from
this
(2)
constants.
323
i
It may
be
seen
that
the
elliptical
integral
is
introduced
(3) --;_cosyl This integral has a real period, which is the integral and 2_, if ICI > I_], and two times the integral taken
if
ICl
Fo.
This conclusion ciently small. What
No.
Let 275.
is
the
still
value
holds
of
_0
in
the
for
us again consider the The exponent a which
general
which
notation appears
this
case,
provided
conclusion
that
would
no
_0
is
suffi-
longer
given in No. 361, which is that there may be developed in powers
hold? of of
327
k.J
the product AA'. It may be reduced to the characteristic
exponent for AA' = 0.
Since we assumethat the solution of the first type is stahle and ]330 is imaginary, A and A' are imaginary and conjugate, and the product AA' is positive. For small values of _, _ decreases when AA' increases. If the reverse were true, solutions of the second type would exist only for P < PO" The crease tive
328
desired
when of
AA'
_ with
value
of
increases. respect
_0
is It
to AA'.
therefore is
therefore
that
for
that
which
which
_ ceases cancels
the
to
de-
deriva-
CHAPTER XXXI PROPERTIES OF SOLUTIONS OF THESECOND TYPE
Solutions
tions
of
371. I cannot of the second
the
Second
pass type
Type
and
the
Principle
over the relationships and the principle of
of
Least
Action
between the theory of least action in silence.
wrote Chapter XXIX just for these relationships. However, in understand them some preliminary remarks are still necessary. Let us assume two variables of the first of a point in a plane. equations will comprise
/331
order
soluI
to
degrees of freedom. Let x I and x 2 be the two series, which may be regarded as the coordinates The plane curves which satisfy our differential 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 n-_ kinetic focus of M on the trajectory (T). This trajectory will touch the envelope of kinetic focuses. I would focus of the order n, is the
E at the point F, accordin_ to the definition like to recall that the tr_ focus of M, or its tn_-h point of intersection of T with the in-
finitely
passing
adjacent
trajectory
of this contact may vary. the curve E, and that the general case.
It may contact
through
happen is of
M.
However,
the
conditions
that F is not a singular point of the first order. This is the most
Let x,= ?(x,)
be the equations of the trajectory (T) and very close, emanating from the point M.
of
a trajectory
Let z I and z 2 be the coordinates of the point be the coordinates of F. Since (T) passes through passes through M, we shall have =,= ?(=,), Due function jectories
ul= _u,),
which
is
M, and let u I and u 2 /332 M and F, and since (T')
_(_,) =o
to the fact that the trajectory (T') is very close to (T), the _ will be very small. I may call _ the angle at which two traintersect the point M. It is this angle which will define the
trajectory (T'), and the function _ will depend on the angle be very small if, as we have assumed, this angle _ is itself and it will vanish with _.
The
(T')
value
of
_'(z2)
(designating
the
derivative
of
_ by
_. It will very small,
4')
will
have
329
the samesign as _. With respect
to _'(u 2) [if we assume that _ is very coordinates has heen defined in such a way uniform, which is always possible], it has
small and if the that the function
system _(x2)
of is
the
_, if
F is
same
sign
sign
if
F is
One that
as
a focus
of odd
characteristic
_(u2) For If
is
of
the
same
let
sign
of
us
the
case
order
as
assume
_ is
of even
order,
in which _2,
that
such
that
and
we
_(u2)
we
pass
If
the
from sign
M of
at a point F' which touches E after F', F',
the
action
The
is
results
to
F'
_ is
proceeding such
sect
tact
(T) close
greater would
to F
In
this
case,
It
cannot
is
of a higher
that
along
be
and we
happen
the
opposite
the
may
say
that
F is than
_(x z)
in
the
fact
close that
the to u2
trajectory
the point F, and > z2)In this
is negative,
(T) intersects
(T)
than
it
is
along
(T').
if
_(u2)
were
negative.
How-
adjacent to (T) there are F, and others which inter-
of F.
that
order
is
sign.
from M than F. In this case, (T') E before F'. When we pass from M to
opposite
short
same
(T).
the trajectories (T') close to F and beyond just
the
(T) touches E after F'. According action is larger (at least in absoproceeding along (T') than it is
_'(u2)
(T)
just
interested
of
is positive,
along
is farther away and (T) touches
ever, in any case, among some which intersect (T)
it has
is positive.
_'(u2)
case, (T') touches E before F', while to a well-known line of reasoning, the lute motion) when we pass from M to F'
are
always
(T') will intersect (T) at a point F' which is not as far away from M as the point F (assuming
when
and
order.
of
the
example,
a focus
F is
an
an
ordinary
the
ordinary point
focus. of E,
and
that
the
con-
first. _333
would
Let
us
develop
The
condition
under
powers
which
there
of _,
would
and
be
let
us
set
a contact
of
higher
order
be +_(,,,) = o
But
we
already
have
and the function _l(X2) satisfies a linear, differential equation of the second order, whose coefficients are finite and given functions of x 2. The coefficient of the second derivative is reduced to unity.
330
If the integral _l(X2) vanishes, for x 2 = u2, it would be identically Therefore,
there
is never
a
as well as its first zero, which is absurd.
contact
of higher
derivative,
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.
the
In one pointed
has the true i_
and the other focus has the
opposite the case
case, _(u2) is on the order of _3. In this case, sign of _, if P is a focus of odd order, and it
sign of _ if of a taloned
F is a focus focus.
of
even
order.
The
In the case of a pointed focus, all the trajectories (T') (T) at a point F' which is close to F and beyond F. Proceeding to F', the action is greater along (T) than it is along (T').
opposite
is
intersect from M
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 more complicated than singular focus.
be an
the case ordinary
that F is a singular cusp. I would then
point of E which is state that it is a
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 passage _(u2) must be of the order _4. 372. correspond ponent and successive
Let
us
of
Let us now consider an arbitrary periodic solution. It will to a closed trajectory (T). Let _ be the characteristic exT be the period. In Chapter XXIX we saw how to determine kinetic focuses (No. 347).
assume
that
whose numerator is p. in No. 347 shows that
_ equals
2in_ T , where
n
is a
commensurable
number
In this case, the application of the rule given each point of 6) coincides with its 2p -_ focus.
331
/334
If, just as in No. 347, we take a unit of time such that the period T equals 2_, we have _ = in. If we designate the value of the function are the values of this at the point M by t0, and if TI, _2, "''' T2p function • at the first, second, ..., up to the 2p_ focus of M, according to the rule given in No. 347, we shall have the following it.
If p ple
of
is
2_,
the
i.e.,
2it.
numerator that
M
of n,
and
its
it 2p_
_pir.
can
be
seen
focus
that
2p_
T2p
-
r0 is
a multi-
coincide.
The trajectory emanating from the point M which to (T) will therefore pass through the point M again around the closed trajectory (T) k + 2 times, if k ÷
is infinitely close after having gone 2 is the denominator
of n.
know
The point M is therefore what category of focuses
classification
presented
in
its 2p _ focus. However, we may wish it belongs to, from the point of view the
preceding
Let us coordinates,
adopt a system of coordinates so that the equation for the
and
_ varies
so
that
jectory.
The
from
curves
0 to
p = const,
2_ when are
the
section. which are similar closed trajectory
one
then
to of
passes
closed
around curves
to the polar (T) is
this which
closed form
an
traen-
/335
velope around each other in the same way as concentric circles. The curves _ = const, form a bundle of divergent curves which intersect all the curves p = const., in such a way that the curve m = a + 2_ coincides with
the Then
parture regarded
curve
_ =
let
_0 be
M. The as the
a. the
value
of m which
corresponds
to
the
point
value of _ which will correspond to this same 2_ focus of the point of departure, will be
point
of
deM,
Let
be the through in the the
equation M. The preceding
sign
of a trajectory (T') which is close to (T) and passes function _(m) will correspond to the function _(x2) given section. We shall have _(m0) = 0, and we must now discuss
of
_[_o÷
332
2_ + _)_].
Wemust therefore formulate the function _(_), and for this purpose we need only apply the principles of Chapter VII, or the principles given in No. 274. For example, if we apply the latter principles, we shall obtain the following. The function @(_) may be developed in powers of the two quantities Jt ¢=%
The
coefficients
of
the
A'e-_oJ
expansion
are
period 2_; A and A' are two integration it is a constant which may be developed _ :_o-_-:_I(AA')_-
equals
The term in.
If
(T')
_0 equals
differs
the
very
periodic
constants. in powers
of
the
respect product
to s, AA'
=_(AA'): ÷ ....
characteristic
little
functions With of the
from
exponent
(T),
the
of
two
(T),
i.e.,
constants
it
A and
A'
are
very small. They are on the order of the angle which I called _ 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', _(_) will be reduced to a polynomial of the third order with respect to these two constants, and I may then write +(_)=
Ae_
+ A'e-_'+f(Ae=%
A'e-_)
where f is a whole polynomial with respect to Ae _, and A'e -s_ cludes terms of the second and third degree. The coefficients polynomial f, just the same as o and o', are periodic functions period 2_.
Under
this
assumption,
order, and since we may write the respect to A and
since
it equals following, A':
s equals
s0,
up
to terms
s 0 + s I (AA') up to terms of neglecting all terms of the
of
only inof the of the
the
second
the fourth order, fourth order with
or
_(_)
--- A e=0,0_ + A'e-=,o _' + at c0AA'(A e=0oe -- A'e-=_o,';
When o and k +
o',
2 for
_
increases
do the
not
by change.
denominator.
(2k
+ 4)_,
The
same
the
Ae=_,
A'e-=0_',
A'e-=0o).
coefficients
holds
Therefore,
+ f(Ae=o%
true
the
for
same
f(Ae_%
of e s0_,
still
f,
as
since
holds
well as s0 -_- = n has
true
for
A'e-_).
333
kj
We
finally
have + (to+ _,_ + _ _)- +(_) = (_k + _)_, _'(A e_0_ --:X'e-_,_:').
However,
_(w0)
is
zero.
The
term
whose
sign
we
must
determine
is
therefore (_£ + 2)_, I shall
employ
I should
o 0 and
first
point
according
to the
in general that this
be pointed term always
AA'(A e_o_'o_o-- A'e-_#o_'
o_0 to designate out
preceding
that
section,
focuses has the
this
the
term
values
is
indicates
or taloned same sign,
o).
of
the
to us
focuses. and that
of o and third
that
It its
o'
for
order
our
which,
focuses
may now be coefficient
w = m0. /337
will
stated cannot
vanish. The
two
constants
A
and
A'
are
related
by
the
following
relationship
q_(_Oo)=o which ties
may
be
written
as
follows,
since
A and
A e_o_0,0 + A'e-=o_o,g In
addition,
conjugate.
A'
a0
The
The cannot In
same
is
purely
holds
product be zero
AA' at
is the
addition,
we
imaginary,
true
for
and
A and
are
infinitely
small
--- o.
(1) o0
and
o' 0 are
imaginary
and
cannot
vanish,
since
the
However, trajectories
equations
these close
(i)
and
A and
have (2)
A e_o_o_0 -- A' e-_0_o _ ----" o, because
quanti-
A'.
therefore positive, same time.
cannot
A'
and
(2) would
entail
equations are impossible. to (T) would pass through
the
the
following
They would point M,
mean which
that is
all clearly
false. Therefore, our term _(_0 + 2k_ + 4_) focuses are therefore all pointed focuses_ Everything,
usual
373. case
k + 2 would
334
depends
on
the
sign
of
always has the same or are all taloned
sign. Our focuses.
_I"
We have disregarded the case in which _i would be zero, an unin which all the focuses would be singular, and that in which equal
2,
3,
or
4.
Following
is
the
reason
for
this.
Wesaw in the computations performed in Chapter VII that the following small divisors are introduced
(see No. 104, Volume I, page 338). The calculation divisors vanishes.
is finished,
and secular terms occur if
one of these
It may be readily stated that if k + 2 equals 2, 3, or 4, we are /338 thus finished with the calculation of terms of the first three orders, which are those which we had to take into account. If, on the other hand, k + 2 > 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 _. It is apparent that the upper limit, which must depends upon the position of M on (T). However, there is not a singular focus. It is therefore 6 equal to the smallest shall assume that 6 is
value which a constant.
this
upper
not exceed the distance MM', it never vanishes, since sufficient for me to set limit
can
take
on,
and
I
Therefore, if the distance MM' is smaller than 6, we may draw a trajectory (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 X = 0, the closed trajectory (T) has _0 = _n 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.
2p_
For X $ 0, the point M will no focus will be located a little
longer be its own 2p_ short of M for X > 0,
focus. Its and beyond M
335
for
%
< 0.
Let
F be
on the position this distance.
this
focus.
of M on (T). It is apparent
The
%, and that it will vanish with the focus F is always beyond M, to the principles teristic exponent. Let
F' be
given The
%. or
MF
will
naturally
We should point out, always a little short
in No. 347, distance MF
a point
distance
depend
I shall designate E as the largest value of that c will be a continuous function of
located
a
depending can never
little
that for % _ O, of it, according
on the value vanish.
beyond
F.
We
may
of
the
charac-
connect
M with
F' by a trajectory (T'), provided that the distance FF' is less than certain quantity 6'. It is apparent that 6' is a continuous function %,
and
that
Let
it may
us
play the provided
set
role that
be
reduced
% > O,
in
to 6 for
such
a way
a of
% = O.
that
M
is beyond
F.
We
may
have
of F', and we may connect M to itself by a trajectory the distance MF is smaller than 6', or provided that
M
(T'),
< 6'. For enough
% =
so We
deviating secting
0,
that
may
e is the
then
2p +
and
inequality
connect
a little (T)
zero,
from
_' = 6 > O. is
the
we
may
take
% small
satisfied.
point
(+),
Therefore,
M
passing
to
itself
around
through
(T) k +
a trajectory
2 times,
and
(T')
inter-
1 times.
.C,
111
pf'/
O,
D
Figure
In
the
figure,
BA
represents
MC is an arc of (T') trajectory bordering
starting upon M.
the
traversed.
trajectories
are
an
12
arc
from M and The arrows
of
(T)
on which
DM is another indicate the
M
is
located.
arc of this same direction in which
The point M may also be connected to itself not through one trajectory, but through two (T'). For one, as the figure indicates, the angle CMA 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 sameat the point of departure as it is at the point of arrival, so that the arcs MCand DMdo not join each other. The trajectory (T'), thus proceeding from M to Mwith 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 would coincide, and this cannot happen according to the principles presented in No. 347.
loops
Therefore, if all of the focuses for % > 0, and we have no more If
all
repeated. that there
the
focuses
were
are pointed, for % < O.
taloned,
We would find that there are no more for % > 0.
the are
same
two
we
line
series
have
two
series
of reasoning of
loops
for
of
could %
be
< 0,
and
375. Let us consider one of the series of loops defined in the preceding 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 corresponds to a periodic solution of the second type. For
example,
let
us
assume
that
the minimum of the action, and that BMD, just as in the figure. Let us
the
trajectory
the angle then take
(T')
corresponds
CMA is larger a point M 1 to
to
than the the left
angle of
M and infinitely close to M, and let us construct a loop (T_) which differs by an infinitely small amount from the loop (T'), having its hooked
point From
M
at M I. and
According point
M
up
to
from to
the
Let
MIC 1 and
M 1 I may
MID 1 be
draw
two
a well-known
theorem,
point
equal
Q will
two
normals
the
the
arcs MP
action
action
of and
this MIQ
along
along
/341
loop. on MIC 1 and
(T')
(Ti)
from
from
the
MD.
the point
337
4
P
to
M1.
We
shall
therefore
have
action (T',)=: action (T') ÷ action(M,P)--
action(MQ)
or action or
(T_ _ = action
(T')
_- action
(,_13lt)(cosC3lA--
cosBMQ),
finally action(T_)
0; OP 3 and oP 4 will for the angles PIOP2,
stability
depends
pass
OPI,
will
functional
then
the origin, and the line p = We shall therefore have two
sign
OP4. OPI and OP 2 will correspond to P0 uo < 0. The function I will be positive negative for the angles P2OP3, PiOP4. We have just seen d_ d--_2. For example,
coincides
zero.
point
of the curve intersect the curve at two points.
solutions,
words
The two branches of the curve determine four the origin. In two of these regions which are positive; in the other two regions, it will be
Let origin.
to
e-_T
_2.
Therefore, one disappear is always
2.
I)=: 2 -- eat-
be
over
positive,
It will also be stable over OP 2 or OP 3.
on
the
_ will and
when
sign
change
the we
correspond to P3OPq, and
of from
solution
pass
the
over
deriva-
negative
will OP4,
be and
343
k.J
The periodic solutions an analytical sequence with which
are
which
are We
correspond respect to
thus
have
and
two
at
analytical
this
series
instant
of
to OP 2 and which are unstable are those which correspond to OP4, and
(5) may However,
have no
at
the
matter
of
time
We have just studied the two of other cases corresponding
curve
periodic
the
two
encounters
in
which
exchange
simplest cases, but there to different singularities
coincide
their
may be which
stabil-
a multithe /348
origin.
what
them
solutions
series
these
singularities
may
an even p + q number of half-branches emanating for _ > 0 and q for _ < 0. Let us assume that origin
they form OP 3 and
stable.
_ = 0,
tude
and to
unstable.
Conversely, those which analytical sequence with
the
for
corresponding to OP 1 are stable, respect to those which correspond
the
following
be,
we
shall
observe
from the origin, i.e., p a small circle about the
order
Let (6)
be
those
which
correspond
to p > 0 and
let (7)
OPl,,._, be
those
which
Then stable may
the
correspond half-branches
solutions
state The
In
that same
and
holds
addition,
(6) will
OPp the
and
for
the
holds
Therefore, let p' and p" be of unstable half-branches
correspond
are
both
true
the for
alternately
For
purposes
alternately
half-branches
OPp+ 1 are
same
OPp+q
solutions.
half-branches true
...,
< 0.
to unstable
these
Consequently,
number
to _
01"¢,+,.,
number p > 0,
of
stable
periodic
brevity,
I
or unstable.
(7).
stable
for
to
or both
OPp+q
of so
and
unstable.
OP I.
stable half-branches that we have
and
the
p'+p"=p. Let q' and q" be the corresponding numbers for _ < 0, so There are therefore only three possible hypotheses
344
that
q' +
q" = q.
p'=
p",
q'= g',
p' ----I a" :,-I,
rl ' = q" -+ _,
/=/--I,
c/= q'--I.
In any case, we have p" _p" _--q'_
in
Let us assume such a way that
that p does not a certain number
q,.
equal q, and, for example, that p > q, of periodic solutions disappears when
we pass from p > 0 to _ < 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 w_ have an analytical series of periodic solutions and that, for p = 0, we pass from stability to instability, or viceversa (in such a way that the exponent _ vanishes). Then q' and p" (for example) are at least equal to i. 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 _ = 0. Therefore, if, for a certain value of p, a periodic solution loses stability or acquires it (in such a way that the exponent _ is zero) it will coincide with another periodic solution, with which it will have exchanged its stability.
379. Let us now return to the case -- that in which the time does not enter consequently, we have the energy integral two degrees of freedom.
and
I shall pursue I shall assume
the solution which very little from T following equations
the that
which I was first going to discuss explicitly in the equations, where, F = C, where finally there are
same line of reasoning as was the case in No. 317_ the period of the periodic solution, which is T for
corresponds for adjacent
_, = o.
to p = 0, periodic
+, = o,
fli = 0, equals T + T, and solutions. I shall write
_., --o,
F = Co,
differs the
i% = o.
(i) which
include
the
following
variables _"
_"
_,, _,
s, _.
According to our hypotheses, the functional determinant of the _'s with respect to the _'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
fll = 0 in
equations
(i),
and
let
us
consider
the
345
functional
determinant A of
/3SO
with respect to
This general
determinant
the
Let
minors
us
_'s
2. No.
64
The
longer
at
the
The same
not
position
of
of
the
then
For
example,
_'s
two
first
zero
of
the
not
all
all
this
orders
at
shall
but
in
F
and to
could
of
same
of
two
two
of
only
the
of
the
the four
happen
if
determinants
of
time,
which
does
same
time.
We
the
not
assume.
zero
along
vanish,
vanish.
determinants
F were
of
The
the
_'s
following
to
at
the
the
periodic
solution.
saw We
in
shall
No.
assume always
eliminate
and
of
F with
respect
to
T were
all
zero
values
a periodic (see
longer
We may therefore A are not zero. us
not
and
this.
equilibrium
Let
will
_'s
determinants
we
zero
derivatives
no
the
of be
_'s,
to _, and with respect zero at the same time? of
which
must
time.
shall
order
T's were
and
correspond of
We
to
the
functional
derivatives they
assume
3.
the
minors
general,
that
no
would
the
respect
in
first
to the theory were true:
All
with
occur,
the
when
_ with respect Can they all be
According following i.
of
consider
four functions variables B.
the
vanishes
solution
strictly
speaking,
but
to a
68).
this. assume
four
that
of our
all
the
unknowns
minors
B and
of
the
• among
first
the
order
equations
(i).
equation
Due
to
of the arrive
346
of
the
the
fact
let
us eliminate
following
form
that
equation
this
preceding section, it will at the same results:
BI,
B3,
has
exactly
be
handled
B4,
r; we
the in
the
shall
same same
still
form way,
have
an
as equation and
we
shall
(5)
i.
When
number,
and
2.
periodic as
When
many
solutions stable
a periodic
continuously
(in
380.
Due
to
Let
the
us
fact
a way
none
after
having
solutions,
loses
that
passage it.
proceed
that
disappear unstable
solution
such
that at the moment of period coincides with
as
or
acquires
a vanishes),
another
real,
the
second
case,
may
of
the
characteristic
when
always solution
that
in
an
even
we
vary
disappear.
stability
we periodic
to
coincided,
always
he
certain
of
the
3_
same
which
exponents
vanishes
for
= o.
except T which
the two which are always coincides with the first
On there cide
shall
the
are with
the
What
may
have For
ber T.
have them, these
other
hand,
periodic
a U
we
of unstable Let q' and
Let the
say
p'
and
solutions q" be the
I may
p"
the
all the Applying
with gated
find the in
if that
we these
refer four
three hypotheses. No. 335.
to
the
numbers The
_
For
of
0.
_
> 0,
become
stable
Chapter
2T,
=
for
solutions
period
XXVIII,
which
coin-
example,
unstable
2T, without for _ < 0.
of
and
having
we
for
_
the
num-
the
< 0.
period
following
three
hypotheses
regarding
q'=q"+1, P",
a + p' = p",
However,
for
in
period
will
the period numbers
the
p'-'-l---p',
shall
of
T
T which number
solution
solutions of period 2T, whether they have the principles presented in No. 378 to
postulate
P'=
is
periodic
presented
type,
stability?
period
be
no
principles
period
which have corresponding
us then consider period T or not.
I find that four numbers:
of
is
second
their
solution
let
the
whose
regarding
there
to
of
solution
stable
> 0,
according
solutions
given
zero, for
q'=
q"-+- 2,
q" = q".
principles cannot simplest
given take and
all most
in
Chapter
values frequent
XXVIII,
which cases
are are
we compatible investi-
347
k_j
Application 381. tain
In Volume
periodic
XXI
solutions
No. 9, and considers which he attributes This fictitious small perturbed He
has
thus
M. planets Jupiter. unstable tic
Darwin
Acts
in
the
Orbits
He
M.
discusses
G.
Darwin
studied
hypotheses
cer-
given
planet which he calls Jupiter, is ten times smaller than that
in
and to of the Sun.
describes a circular orbit around the Sun, and a having zero mass moves in the plane of this orbit.
acknowledged
the
existence
of
certain
periodic
solutions
included in those which I have called solutions of the which he has studied in detail. These orbits are referred turning around the Sun with the same angular velocity as orbits are closed curves, in relative motion with respect axes. has
called
the
first
class
of
periodic
orbits
the
class
of
The orbit is a closed curve encircling the Sun, but not encircling The orbit is stable when the Jacobi constant is larger than 39, and in the opposite case. The instability corresponds to a characteris-
exponent For
The
having-_-as
values
of
periodic
the
the Jacobi
solutions
corresponding
imaginary constant of
orbit
the
will
be
part. which
second a
are type
closed
passing around the Sun twice. The two loops from each other, and both differ very little We a later M. are
shall
study
these
solutions
of
the
close whose
curve
39,
with
there is
second
type
are
double.
a double
of this curve differ from a circle.
Darwin also those which he
obtained oscillating we discussed in No.
in greater
obtained
satellites
satellites which he 52. They are always
which,
strictly
to the system of moving axes under consideration, circling Jupiter, but not encircling the Sun. For is
to
period
point very
little
detail
at
point.
Finally,
which
H.
the
A.
therefore
and
/352
of Darwin
Mathematics,
detail.
a perturbing a mass which
planet planet
which are again first type, and to moving axes, Jupiter. These to these moving
of
to
C = 40 stable.
(C is For
the Jacobi constant), C = 39.5,the satellite
called a and unstable.
speaking,
describe
with
closed
b,
respect
curves
en/353
we have only one satellite A A becomes unstable with a real
exponent _. 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 _. For C = 39, we obtain the same result. For C = 38.5, the satellite C becomes unstable for
348
C =
with 38, we
a complex obtain
exponent the
same
_
(whose
result.
imaginary
part
is _).
Finally,
Wemust therefore consider three passages: i.
The passage of satellite
A from stability
2.
The appearance of the satellites
3.
The passage of satellite
B and C;
C from stability
The last two passages do not entail
to instability;
to instability;
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 _ is real for the unstable solution. This conforms with the conclusions reached in No. 378. The passage of the satellite C from stability to instability no longer presents any difficulties, because the exponent _ is complex in the case of instability. The conditions presented in No. 380 therefore hold. We therefore 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 _ is real in the case of instability. According to No. 378, we should therefore have exchange of stability, with other periodic passing around Jupiter only once. calculations of Darwin.
by A.
We are naturally led Darwin do not represent
Other
The unstable
considerations
to think that the unstable the analytical extension
lead
stable satellites satellites A have
solutions corresponding This would not seem to
to
A have orbits
the
same
to closed curves result from the
satellites A discovered of the stable satellites
result.
ordinary closed curves for orbits; in the form of a figure eight.
the
/354
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 i, Darwin gives the curves of zero velocity. For C > 40, 18, these curves intersect the axis of the x's between 18, and the passage
but
the Sun occurs
We are left with this is no longer
ponding
to
C =
40
and
and Jupiter, but this no longer between C = 40 and C = 39.5.
the hypothesis satisfactory. to
C = 39.5.
that the cusp Let us compare The
first
holds
for
C < 40,
is located beyond Jupiter, the two orbits corres-
intersects
the
axis
of
the
x's
349
k_J
twice at a right angle, once beyond Jupiter and once just short of it. Let P and Q be the two intersection points. In the same way, the second orbit (if we disregard the double point) intersects the axis of the x's twice at a right angle, once beyond Jupiter, and once just short of Jupiter. Let P' and Q' be the two intersection section point P or P' which is beyond sign
of dd-_t • We
other.
shall
However,
see
that
this
dd-_t would
have
to
points. Jupiter, sign
Let and
us consider the let us determine
is positive
change
sign
when
for
one
passing
interthe
orbit
or
through
the
the
cusp. The
point
P,
the
hypothetical
cusp,
and
the
point
P'
cannot
therefore
be regarded as the analytical extension of each other. We must then assume that at a given moment an exchange has occurred between the two intersection points of the orbit of the satellite A' and of the x-axis, that which is located on the right passing to the left, and vice versa. Nothing in the behavior of the curves constructed by M. Darwin justifies such an assumption. Therefore, analytical lites
I may
extension
A become
conclude of
the
that
stable
the
unstable
satellites
satellites
A.
But
A are
when
do
the
not
the
satel-
stable?
I can only formulate hypotheses on this point and, in order to do /355 otherwise, it would be necessary to reconsider the mechanical quadratures of M. Darwin. However, if we examine the behavior of the curves, it appears that at a certain time the orbit of the satellite A must pass through Jupiter, and that it then becomes what M. Darwin has called an oscillatin_ satellite. 383. these
Let
planets The
us
study
from
orbits
the
planets
stability
of
these
to
A in
greater
detail,
and
the
passage
of
instability.
planets
correspond
to what
we
have
designated
as
periodic solutions of the first_The orbit with a double point, which passes around the Sun twice and which differs very little from that of the planet A at the moment when the orbit of this planet has just become unstable' corresponds to which we have designated as periodic solutions of the
second If we
type apply
(47). the
procedure
second type from those shall obtain solutions In
solutions
of
of of
the
the the second
by
which
we
deduced
periodic
first type to solutions second type exactly. type,
the
mean
of
anomalistic
solutions
the
first
motions,
of
type,
the we
which
differ ver_ little from the mean motions strictly speaking, are in a commensurable ratio, 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 close to a simple commensurablenumber. Since the orbit the Sun twice, this ratio will be close to a multiple of
meanmotions is must pass around i _.
In other words, at the momentof passage, the term which M. Darwin has called nT must be close to a multiple of 7. In effect, this is what occurs. with the following c=
of
be is
The mean Jupiter.
A stable,
nT .= r54°,
A stable,
nT = 165°,
39
A unstable,nT
= ,77%
A unstable_nT
---w9_o
seen that the passage close to 180 ° . motion
of
of M.
_o
C = 38,5
It can number
tables
C=39,5 C=
this
The
the
planet
must
A is
be
made
therefore
Darwin
provide
around
nT
=
170 °,
almost
three
times
us
and
that
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.
set
384. Let us the following,
again just
consider the notation as in this section z,=
L--G,
Fo =
The term L must have eccentricity must be of the eccentricity,
their
shall
313,
and
let
us
_Y2---- l'+g--t,
R+G=
Fo+/tF,+...,
_
x, -- x,
the same sign as G (see page 201, in fgne), and the very small. Since x I is on the order of the square this variable will also be very small.
Since we only wish to determine stability, we shall be content We
in No.
x,=L+G,
2yi---- l--g4-t, F'=
given
therefore
neglect
the number of periodic with an approximation.
_2F 2 and
_FI, we only take into account secular period, and we shall neglect the powers shall have
the
following
terms.
solutions
and
In
term
terms and terms with a very which are higher than x I.
the
long We
351
F" I
where
a, b,
c are
functions
term
which
has
period
with
The
very
2y2
- Yl-
We
long
then
been
we
may
-F"
CXi
COsuJt
of x2,
and where
CXlCOS_
is
the
very
long
retained. terms
therefore
are
terms
with
_ +
3g
- 3t,
i.e.,
terms
have
have F'=
and
b.z'l
only
period
We
_-= _t-+-
apply
(xt+
a
the method
2 -+ :'1,, ......
x,)'
Xt' I
+_.(a_a_bxt+cxlcosw
",
)
of Delaunay. _357
The
from
canonical
which
we
equations
the
the
integral
have F'=
With
have
approximation
+ - -- -- + 1_(a + bx,+ • a
(/, = xl)' which
has
been
adopted,
cxtcos_).
we
may
replace
a,
b,
c by
t ao--
designating
that
place
k.
x 2 by
which
a,
the
coordinates
us
t
constants
which
_ = bo-- 2ao, depend
assume of
that
a point
C designates This
double
352
curve
point,
ao,
a'o, bo,
co when
we
re-
k is in
also
this
we
have
a constant, and
,!x_cos!_,_!_ sin_ let
us
compile
the
are
rectangular
curve
C_
constant.
depends
double
and
3,r, + _I(_+ _xt + 7x, cos_). ".
k ' 2
a plane,
a second
-_ = co
on k,
F r =
where
by
Thus,
_. F' = ....... tk--x,)' Let
Cot
bo_
da b c, become dx---_' '
_t--_ao, designate
2xlao_
on
point
the will
two
constants
correspond
k to
and
C.
a periodic
If
it has solution,
a
which will be stable if the two tangents to the double point are imaginary, and unstable if the two tangents are real. Weshould note that the curve is symmetrical with respect to the two axes of the coordinates and that the two douhle points, which are symmetrical 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 coordinates, so that they will be obtained by setting _0_0_
If we set
W=_.
k
the curve F' gents to the
= C passes through the origin double point are given by the 3 .....
f,':_
Therefore,
a
:,-
l_
-q--
and has equation
_y costo
_
a double
point.
The
tan-
/358
O.
if 3
the
tangents
are
imaginary.
If 3+_.
the
tangents
are
real.
Finally,
(2)
__y,
if 3
--sT>_ the
tangents
are
again
(3)
---._÷s_,
imaginary.
The coefficient B is positive. also assuming that y is positive. If change _ into m + _.
I wrote the preceding inequalities y were negative, we would only have
to
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 (i) or (3) hold, and is unstable when the inequalities Let = 0.
(2) hold. us now
study
the
double
points
which
may
be
located
on
the
line
353
kJ
If
we
set
_
=
0,
the
function
F'
becomes
F' = --_'--+ ....._, +,_ ( h -- x, )' "_ 2 Keeping
k
minima
of
constant, F'
are
if given
we
vary
by
the
x I
from
0
to
(4)
+ I1x,(_ ÷ _) = C.
k,
we
find
that
the
maxima
and
equation 3 .... ( k -- x l )'J
which
has
tion
in
a the
solution
if
opposite
case.
Therefore, stantly
decreasing
a maximum,
and
This or
if
if then
maximum
rather
to
the
the
inequality
inequality it
(3)
holds.
The
(3)
holds,
does
not
and
does
hold,
function
F'
not
the
first
have
function
a
F'
increases,
solu-
is
con-
reaching
decreases. corresponds
two
(5) + _(_ +-()-- o,
",
double
to
a
points
which
determine
how
double are
point
located
symmetrical
on
with
the
line
respect
_ = to
0,/359
the
origln. However, given tion
value of
k.
we
the
constant
We
must
deduce
However,
from
which
must
of
equations
we
C.
x I from
(4)
we
may
Equation
and
it
obtain (5)
as
a
function
(5)
may
be
written
F'=
C,
dF' dxl
-- o
dC
dF'
dk
_
dk
dxL
dF' +
- _ d _1r'
d IF'
terms
containing
B,
dxl
we
we
dF'
d' F'
354
+
_-=
--|
dx,
have
dkdxl
and
O.
have
_--
which
7ix,'
-_!
dF' _
d],"
_
dk --
(lar--_l÷tlk
from
of
dF'
d_ F' -- d--_
ct_ F' =°;
dx_
_
t '_. -- (k--x,)
double us
have dx,
Neglecting
these
provides
_
C.
with
points x I as
for a
func-
a
d/c dxj It
results
from
this
that
x]
dC
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 _ = 0. Let
C O be
the
value
of
C which
satisfies
the
double
equality
point
2 k Co=_+--+iI_t, ._ ___ /c _
We
shall
see
= 0 and
on
that,
that,
for
for
C > CO,
C < CO,
The same discussion the line _ = _. The
3 _ -_- i._(_ -__7)_
there
there
has If
a solution
C 1 is
the
if
value
be
not
be
a double
two
of
them.
on
the of
(2) or
satisfies
the
line
located
3 :_ + ,_(_ - "c)= o
inequalities C which
the
f360
may be applied to the case of double points values of x I will be given by the equation
(h---_ x,--3_which
will
will
o
(5') (3) hold. double
equality
k c L= %, + 7 + sx' /._ -- _.+ s(_ --7)=0, the
is
for the
condition
for which
there
are
two
double
points
on
the
line
C < C1 . We would like to point out that C 1 > CQ, that C O is the value of C which one passes from inequality (2) to inequality (3), and that C1 is one for which we may pass from inequality (i) to inequality (2).
When compiling the curves, we would readily real for the double points located on m = O, and for the double points located on m = _.
find that
that they
the are
tangents imaginary
are
355
Wemay therefore sum up our results as follows: First
case C > Cl.
The
inequality
The
solution
There
of
is no
Second
(i) holds. the
first
solution
of
type
the
(planet
second
type
A)
is stable.
(orbit
with
double
point).
case ci> C > C_,.
The
inequalities
The
solution
There
is
Third
case
(2)
of
the
hold. first
a solution
of
type
the
becomes
second
type
unstable. which
is stable. /36l
C < Co. Inequality The
(3) holds.
solution
There
are
of
two
the
first
solutions
one of which is unstable. located on the line m = _, located
on
the
These the
value
adopted
38.
356
not
by
are M.
verified
It is therefore had continued
of The and
the
is stable. second
type,
one
of which
is
stable
and
first corresponds to the two double points the second corresponds to the two double points
m : 0.
conclusions
I have
if he
line
type
valid,
Darwin,
this,
provided that _ is 1 _ :_, sufficiently
but
it seems
very
sufficiently
small.
Is
small?
likely.
likely that M. Darwin would have ohtained stable orbits his study of the planets A for values of C smaller than
CHAPTER XXXII PERIODICSOLUTIONS OF THESECOND TYPE 385. Let us again consider the equations of No. 13 ,IF
dxi
dyl
dt - dye' with p degrees these equations by
the
period
The
x's
of freedom. will have T,
integer
the
numbers
However, this is is not zero. The
hessian x.
is
This to recall
zero,
YI'
k I, k_,
...,
_
statements such that, yp
may
F 0 does
what occurs Y3, Y%; Y5,
increase
be
in Y6
not
given in No. 42, when t increases respectively
by
arbitrary. of is
F 0 with invalid
depend
on
respect to when this
all
the
the three-body problem. represent, respectively,
the
variables
I would like the mean
the perihelions and of the nodes, and that variables x I and x 2 which are proportional axes.
us consider a periodic one solution will be
differences one period.
...,
when
longitudes of the planets, of depends only on the two first the square roots of the major Let ventions,
Y2'
(i)
F = F0+ ,_F,+...
only valid if the hessian proof presented in No. 42
particularly
is precisely that Yl, Y2;
dF
-i;F = -- 7/_'
According to the periodic solutions
variables
1362
F0 to
solution. According to the stipulated conassumed to be periodic, provided that the
of the y's increase by multiples of 2_, when t increases In actuality, F only depends on these differences.
by
Let
be
the
quantities
by
which
the
y,--yc, when
t In
increases Chapter
by
following
y=--y_,
/363
increase
y3--)'6,
)'_--y_,
y,--y,,
a period.
llI we
solutions corresponding k3, ky and k 5 are zero.
could to
only
establish
arbitrary
values
the
fact
of k I and
that k2,
there
are
assuming
periodic that
357
It may be inquired whether in this case, lust as in the general case, there are periodic solutions corresponding to arbitrary values of the five integer of the
numbers k, solutions second type.
which
I would
like
to
386. Do these solutions of the second type to answer in the affirmative, based upon reasons the fact that the form of the function F needs to order to obtain canonical equations to which the in
No.
when to
as
solutions
exist? It is a temptation of continuity and considering be modified very little in line of reasoning pursued
42 applies.
However, one difficulty is entailed. we cancel the term we have designated
the
disturbing
If
the
What happens to these solutions as _ and which is proportional
masses?
disturbing
masses
are
zero,
the
two
planets
Kepler. The perihelions and the nodes are fixed, the numbers k 3, k4 and k 5 can have no other value This small, itself
designate
difficulty
the two becomes Let
us
may
be
resolved
as
that
the
two
follows.
planets,
which
the
and it would than zero. If
planets will obey the laws of Kepler, infinitely small at certain times.
assume
obey
are
the
very
are
their
far
of
appear
masses
unless
laws
away
that
infinitely
distance
from
each
other, both describe a Keplerian ellipse. It could happen that these two ellipses will meet, or will pass very close to each other, in such a way that the distance between the two planets becomes very small at a certain time. At this time, their mutual perturbing action could become significant, and the two orbits could undergo large perturbations. The planets, moving /36____4 away from each However, these perihelions I would
other again, new ellipses
and
the
like
nodes
would then describe will differ greatly will
to employ
the
although it is not a collision planets do not come in contact only be rather small in order the smallness of the masses. However
that
may
be,
account, it is no longer are fixed for _ = 0, and
undergo word
Keplerian from the
considerable collision
ellipses again. old ellipses. The
variations.
to designate
this
phenomenon,
in the true sense of the word, since the two and since the difference between them need to have considerable attraction, in spite of
if we
take
these
orbits
with
collisions
into
valid to state that the perihelions and the nodes that consequently the numbers k 3, k4 and k6must be
zero. We must
thus
conclude
that
solutions
of
the
if w_ make _ strive to zero, they will tend to be series of collisions. However, this rough sketch more
358
detailed
examination
is necessary.
second reduced is not
type
exist
to orbits sufficient,
and
that,
with and
a a
387. Let us first consider the effect of the collision. Let E and E' be the ellipses described by the first planet before and after the collision; let Ei and Ef be the ellipses described by the second planet. It is apparent that these four ellipses must intersect at the samepoint, in such a way that the two planets describing these four orbits will pass through the encounter point at the time of the collision. As long as the distance between them is considerable, the two planets describe curves which differ very little from an ellipse. During the very short period of time when the distance between them is very small, they describe orbits which are very different from an ellipse. These orbits may be reduced to small arcs of curves C having a radius of curvature which is very small; these arcs differ very little from arcs of a hyperbola. At the limit, the very short time of the collision may be reduced to an instant. The small arcs C may be reduced to a point, and the orbit, being reduced to two arcs of an ellipse, has a hooked point. In order to define the orbits E, E', El, _i' it is necessary to know the magnitude and direction of the velocities the two planets P and P1 before and after the collision. _at are the relationships between these velocities? I would first like to note that the velocity of the center of gravity of the two bodies P and P_ I must be the samebefore and after the J365 collision. The magnitude and direction must also be the samebefore and after the collision. Wemust now consider the relative
velocity
of P with respect to PI;
the
same
before
the
direction
of
axes whose origin is at PI' in magnitude and direction,
and the
the magnitude of this velocity must be sion, but it may differ in direction.
after
Following is the the collision.
rule
for
Let us consider moving a line AB which represents, city
of P with
through
the
respect
point
PI'
to P1 since
determining
before the
body
the
collision.
which
moves
and
This at
after
this
colli-
velocity
let us consider relative velo-
line
the
the
AB must
velocity
pass
which
it
represents must collide with the point PI, which is fixed with respect to our moving axes. However, this is only valid at the limit. This is only valid because we regard the masses, on the one hand, and the distance at which other
the mutual attraction of P and hand -- i.e., that which could
P1 begins to be manifested, be designated as the radius
as infinitely small quantities. It would therefore be more the distance d from P to the line AB is an infinitely small same order as the radius of action. Let respect and the
A'B'
be
the
line
which
represents
the
to P1 after the collision. In terms distance from P1 to A'B' equals 5.
relative
the action
exact to quantity
velocity
of magnitude,
on of
A'B'
of
--
say that of the
P with
equals
AB,
359
k_1
We
finally
have
the
rule
for
determining
the
direction
point P1 and the two lines AB and A'B' are in the ties which are infinitely small of higher order). A'B'
may
be
determined
proportional
to
It may
are
6 and
thus
be
The only two the following:
magnitude and alone. These The
Let
toward
follows.
to
the
seen
The
square
that
of
the
tangent the
length
direction
and us
try
which
area
to
constants
compile
the
the
solutions
of
like
to point
out
must
AB.
of A'B'
may
not
orbits the
of
of
with
this
be
angle
is
arbitrary.
velocity
be
changed
in magnitude
by
collisions
second
The
(up to quantiof AB and of
upon our four velocities the center of gravity in
permanence of the relative may be given as follows:
the
of A'B'.
plane angle
of half
conditions which must be imposed permanence of the velocity of
direction; conditions
energy
388. limits
as
same The
type
tend
the
which when
collision. are
the
_ strives
/366 to
zero. I would
first
assumed in order that consecutive collisions ellipses described tive collisions.
by the planets P These two ellipses
they have a common section points and Let
us
that
at
least
that
we
are
collisions Let us Let E
and P1 in the interval must intersect at two
focus, they are in the same the focus are on a straight
assume
two
such an orbit may be periodic. never occur at the same point.
dealing
with
plane, line.
of two points
unless
an exception.
must
be
assume that two and E l be the
the
Let
Q
consecuand, since two
inter-
and
Q'
be
the two intersection points of the ellipses E and _i which I shall assume are not in the same plane. These two points are on a straight line with the focus F; let E and E'1 be the ellipses described by the two planets after the collision. They will pass through the point Q, where the collision has just been produced, and they will not be planes will intersect along the line FQ, point (which must exist if two consecutive same point) will be located on this line two ellipses E and El will have the same the points F, Q and Q' are on a straight '
in the same plane in general. Their so that their second intersection collisions never occur at the FQ. I would like to add that the parameter. Due to the fact that line, the inverse of the para1 1
meter
E l will
of
the
Under purposes
this
of
the points
ellipse
where
360
must
of
assumption,
clarity,
let
the
We may specify located on the same We
E or
the
we us
four
ellipse
shall
assume
collisions
employ
four
the
be _+
following
collisions;
two
ellipses
let
procedure. QI,
Q2,
Q3,
For Qh be
occur.
these four points arbitrarily, line passing through F.
construct
_,-
E and
E l which
provided
intersect
that
they
at QI
and
are
Q2,two
ellipses E' and E'I which intersect at Q2 and Q3, two others E" and E'I' which intersect at Q3 and Q_, and finally two others E'" and E'i' which intersect at Q4andQI. The orbit of P is composedof arcs pertaining to the four ellipses E, E', E", _" , and the orbit of P1 is composedof arcs pertaining to the 111
four
ellipses
El,
E'I, E_', E l
We shall specify constants must be the sions
(orbits
intervals. this is the
E
and
.
the energy and area constants same for the interval between
El)
for
the
following
interval,
arbitrarily. the first two and
According to the statements presented in only condition which must be fulfilled.
for
all
These colli-
the
the preceding
/367
other section,
In order to compile E and El, we shall proceed as follows. L_t us consider the motion of three bodies. Since we assume _ = 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 P1 must leave the point Q1 simultaneously in order to arrive at the point Q2 simultaneously. When P and P1 go from Q1 to Q2, the true longitude of P increases by (2m + I)_, and that of PI increases by (2m I + i)_. We may still specify the two integer numbers m and m I 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 principle of Maupertuis, and Maupertuis always has a minimum. We must now area constants. the line FQiQ2.
be resolved. We need only apply the action, which is essentially positive_
determine the planes of the two We therefore know the invariable The areal velocity of the system
vector perpendicular we know. It is the planets, represented equal, respectively,
ellipses. We know the plane which passes through is represented by a
to the invariable plane, whose magnitude and direction geometric sum of the areal velocities of the two by two vectors whose magnitude we know, since they mp and mlp , where m and m I are the masses of the two
planets and p is the common parameter of the two ellipses E and E 1. therefore compile the directions of these two base vectors which are dicular to the plane of E and to the plane of El, respectively.
We may perpen-
v!
The
terms
E'
and
E_,
E"
Let
us
now
assume
and
El,
...,
may
be
determined
in
the
same
way. 389.
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 E l . 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 E l .
361
kJ
Let
us
assume
that
during
the
interval
under
consideration
the
planet
P has performed m complete revolutions, and that the planet P1 has completed ml complete revolutions. We can arbitrarily specify the two whole numbers m and m I. 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. vector which represents the areal velocity of be decomposed an infinite number of ways into
the can
Consequently, we know the system. This vector two base vectors which
represent the areal velocities of P and PIWe shall arbitrarily this decomposition. If we know the two base vectors, we know the planes of the two ellipses and their parameters. The orientation of these ellipses in its plane remains to be determined. We will it
by
passing
the
Summarizing,
of
we
i.
The
point
2.
For
all
3.
For
each
the
areolar
In satisfy
can
Q and the
through
the
arbitrarily the
interval,
Q.
specify:
number
intervals,
point
of each determine
of
the
intervals;
area
the whole
constant
numbers
m
and
the
energy
and m I and
the
constant; decomposition
vector.
order
to make
certain
390.
ellipse
specify
Let
the
problem
inequalities us
tractable,
which
disregard
the
I will
exceptional
these not
arbitrary
numbers
must
describe.
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, Q2, ..., be the points where the successive 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
two planets pass from of the radius vectors vectors,
nor
We know planets
has
the
of
duration
that
in
increased
the
intervals,
for
example,
Q1 to Q2We will arbitrarily FQ 1 and FQ 2, but not the angle of
this by
the
one
where
the
the magnitude these two radius
interval.
interval 2m_.
the
specify between
Let
the us
difference arbitrarily
in
longitude
specify
the
of whole
the
two J369
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 E]. This means that the principle of Maupertuis must be as was done in No.
362
applied. 339 and
However, the the Maupertuis
Hamiltonian action must
action must be defined be derived according
to the procedure of Nos. 336 and 337. Unfortunately, this Maupertuis action is not always positive and therefore one is not certain that it always has a minimum. Summarizing, we can arbitrarily
specify:
i.
The number of intervals
and the lengths FQI, FQ2, ...;
2.
The area constants and the energy constants;
3.
For each interval,
the whole numberm.
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 _ = 0, there are certainly somewhich are planar. It is also possible that there are somewhich are not planar for _ > 0, and only becomeso 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 t Obe a time before the first collision and t I a time between the first and the second collisions. In the sameway, let t 2 be a time between the second and the third collisions. For the discussion I will assumethat there are three collisions. I will call T the period in such a way that at the time t O + T the three bodies appear in the sameconfiguration as was the case at the time t O. 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
orbit is regarded configuration at Let
x_,
0 x2,
and
the
as the
periodic if the three end of the period.
tO +
T.
One
will
0 x H be
...,
t O fo_ the collision Let x_ be the values sion orbit, x$ their
nodes.
the
In
all, there
values
of
are
bodies
these
eleven have
variables. the
variables
same
at
the
The
relative
instant
orbit under discussion and consequently for _ = 0. 3_ of these variables at the time t I for this same collivalues at the time t2, and x$ their values at the time
have ,r_ = x, ° + _ i_,,._
where mi tricities
is a whole number which and the inclinations.
must
be
zero
for
the
major
axes,
the
eccen-
363
kj
Let
us
now
consider
an orbit
which
is slightly
different
from
the
col-
lision orbit. from zero. In
Let us assign a very small value to _, but different Q this new orbit, our variables will have the values x_ + _i 1 1 2 2 to, x i + 8i at the time tI, x i + _i at the time t 2 and finally
at the time 3 3 x i + 8i at the The
tO +
condition
Assuming the time
and
time
for
u = 0, t I, the
T +
_.
which
the
in order variables
solution
is periodic
that a collision _ must satisfy
with
period
occurs between two conditions.
T +
the
T is
time
tO
Let o 0
._I
be
these
two
Let
conditions.
us set 0_.0 _,. - r,,
/,(_0)::_?_,A(_7)=_; it
can
p.
By
be
holds
that
applying
the
for
In that
seen
the
order
that
_ = 0),
two
Let
I
Q's Yi
the
and
then
set
find
that
the
of
U.
The
Finally, t o + TT,
364
two
there
be
of
holomorphic
functions
of
Chapter
it
shown
a collision
conditions
the
us
and
principles
... ii);
II,
can
be
0v 7i s and
the
that
the
of same
8_'s.
Replacing of
0v Bi s are
the
(k=3,4,
B_'s
of _,
in
in
holds
between
necessary,
relationships
and
Iv B i s and same
are
then
setting
the
which
(I) by _ = 0,
their
2v B i s are holomorphic O's and true for the _i '
order
that are
there
be
necessary
a collision which
tl
and
t2
(assuming
write
as
follows
values
as
a function
I obtain
the
conditions
times
I may
I may
"Ivs functions of the Yi consequently for the 8_'s.
between write
the as
times
follows
t 2 and
f,< _i ) =f_(;_,' If and
of
I may
and
2_S 8i
the p,
and
are
if we
replaced then
set
by
their
p = 0,
) -- o.
values
they
as
a function
of
the
Iv Yi s
become
set
I then
find
tions
of
the
tions
of
the
The holomorphic
that
¥i2's and 2w Yi s, of
the of
_.
In
the
_,
and
of
T.
relationships with
Bg'Sl , the
B_I =
respect
to
8_ are the
8
's, same
and way,
therefore
y_'s,
_,
the
and
the
that this is necessary, the orbits traversed in
holomorphlc
B 13's are
whose
T.
equations
These The
func-
holomorphic
equations
discussed in the same manner as in Chapter III. of the second type could then be demonstrated. I do not believe deviate too much from
812, s are
existence
two
of
func-
terms
are
could
be
solutions
because these solutions actuality by celestial
bodies.
365
CHAPTER XXXIII DOUBLY ASYMPTOTIC SOLUTIONS
Different
392.
In
ourselves
to
order
Methods
to study
a very
special
of
Geometric
doubly case,
Representation
asymptotic that
of
solutions,
Section
No.
we
9:
shall
Zero
confine
mass
of
1372
the
perturbed planet; circular orbit of the perturbing planet ; zero inclinations. The three-body problem then has the well-known integral called the Jacobi integral
•
Returning tinguish
between
following
to No.
299
several
devoted
to
cases.
We
this saw
problem
from
on page
159
Eo.
that
9, we
must
dis-
must
have
the
we
inequality
_
__ .....
f*i
ft
,_ _ (_,+ ,,,)= v+ 2
-- (_' 4--_')> --h.
(1)
2
We then distinguished between the in which -h is sufficiently large
case in which m i is much (page 160). We saw that
smaller than the following
m 2, and
curve V ,-n' ($'+ ¢')= -- h
(2)
2
may be broken down into three closed branches which we have called CI, C2 and C3. Therefore, in view of the inequality (i), the point $, n must always remain inside of CI, or always inside of C 2, or always outside of C 3 (_, are the rectangular coordinates of the perturbed planet with respect to the moving
axes). We
for
shall
curve
assume
below
that
the
(2) to be
broken
down
into
point 5, _ always remains inside of the perturbed planet to the central for the distance r I between the two
lated
This hypothesis corresponds to on pages 199 and 200 -- i.e.,
Figure
9, We
and
shall
the
point
employ
the
x I, x2
value
the
closed
constant
-h
branches,
is and
large that
enough the
C2. In this way, the distance r 2 from body may vanish, but this is not true /373 planets. the the
remains
notation
of
three
given
following hypothesis, curve F = C has the on
the
in No.
utilizable 313,
and
which we form shown
arc we
formuin
AB.
shall
introduce
the Keplerian variables L, G, i, 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. Wesaw in No. ii 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. Wethen have [;'.=-Rq-G----
V/_--
:'t -+- G -4- _V/'7-'_t
L' where _ and 1 - U designate central body, r I designates nates the constant and r 2 designates Just
as
It
",
the masses of the perturbing the distance between the two
distance from the perturbing the distance of the perturbed
in No.
313,
we x_::
shall
v-F, =
I would like to stress the function F l always n cannot leave.
L--G,
to the to the
central central
body, body.
x,=L+G,
r Fo --= 7-L--_ +
_, F,,
¢' - ;" ' _-r,: ....
'2 G --
(xt x,)_ +
_ ¢'--:-,'] '......
t_
-.t-
("_-'-
.'_ -- .'r____._w. 2 '
"I)-
the following important point. It can remains finite in the region from which
_'._
of the 1 desig-
2y,=l+g--t;
We shall employ the method shall represent the configuration rectangular coordinates are X =
body body
body and planets,
set
_)'l=l--g+t, I"' = F_,-_-
(r_--I--,'])
2 V'i --
of
representation of the system
cosy,
y
¢.;;--7 __-,- .. ¢;, _o_y, Z=
=
by
_
¢____ _.,
be seen that /374 the point _,
given on page 200 , and we the point in space whose
siny,
_ = ¢., _o_y,
"_¢'_ _iny,
367
xl is constant, the point X, Y, Z It can be seen that, when the ratio x-_ describes a torus. This torus may be reduced to the Z axis when this ratio is infinite, and may be reduced to the circle Z-_-o_
when
this
ratio
The
just
small.
could
become
does
the
would
infinite
function
not for
be
dF 0
the
very
little
hypothesis
which
vanish because the be readily reduced (on page
2
201,
If x 2 is
dF 2 _-x2 can
because Y2
is
always Let
shall
are
We
saw
consi-
on page
201
that,
consequently
in
terms
n2
cannot
constant C (the constant C given in No. constant h given in No. 299) is larger 3 3 set _ instead of _ everywhere).
must
very
only
small,
become
a point
we
shall
infinite
except X,
Y,
therefore
for
Z, such
small
that
Y2
=
313 may than
have
x 2 = 0,
for very
of
from
which
it
follows
that
x 2. 0.
On
the
half-plane
we
have
When
Xl,
x 2, Yl,
to M I -- will The
368
under
except when x I or x 2 is ' dFl dFl which derivatives dy I, dy 2
dF0 d-_2and
Y2
vary
in
X > o. conformance
the point X, Y, Z will describe a certain creases constantly, reaches the value 2w,
given plane
region
a result,
dealing,
Y = o,
moved
the
dF' -- n_= d_,.
we
increasing,
M be
As
the
in
energy to the
we
not
for
dF0 and _2"
from_
with
true
dF' _,
finite
F I itself
r 2 = 0.
---n,=
differ
l,
dF 1 dFl _ and d-_2 remain
as
This
-Y_==
zero.
derivatives
deration, very
is
Xz-i
point
M 1 is
again then
be the
located
on
consequent
with
differential
equations,
trajectory. When Y2, which inthe point X, Y, Z -- which has
the half-plane of M,
according
in No. 305. Since Y2 is always increasing, has a consequent and an antecedent. There
Y = O, to
X > O.
the
definition
every point on the is only an exception
halffor
/375
k.J
very from
small x2 -- i.e., for points on the half-plane the origin, or very close to the Z axis.
ted
We shall have an integral to this word in No. 305.
invariant, Let us try
the
Due to the fact that the equations following integral invariant
which
in terms of to formulate are
canonical
are
very
far
the meaning attributhis invariant. equations,
they
have
_Z,r,_lx_dy, dy2.
Let The
us
set
invariant
x2 z =--Xl , and
will
We may (due
deduce to the
us
F' , z, Yl,
select
Y2
as new
variables.
become
- aF' ,w'-j-._i,,-i+_-n] " ._ (;v'(z: ,ly, 4r, _ ['_ aV'(t_.dr, at,
--
variant
let
the following existence
triple invariant the integral F'
of
/_i
from = C)
this
quadruple
dz +_:_" _'y' d.r, n, ¢_F'
In
this
triple
as
functions
replaced
integral, of
we
z, Yl,
assume Y2
xt
Let nate the become
us now take Jacobian of
us
--
which
,v t z,
of
the
are
d.r,
equations
F' ---- C.
X, Y, Z, and let us employ respect to z, Yl, Y2The
A to desig/376 invariant will
g
_n
it
follows
again
set
_ sinr,
2co,.r,'
Z= ¢t;+ _- 2cos.r_'
that X ::
us
means
dF'
n2--
set
_-
Let
that x,,x2,n,=--_zT_,
x_ dX dYdZ (xtnt+x,n,)_
R
from
by
the variables X, Y, Z, with
f
Let
in-
D -= [(R
R cosy',
-- ,)'
Y ::- R sinyt.
+ Z'][(R
+ ,),-t-
Z']-
369
k_J
A simple calculation
Our invariant
provides the following RD may therefore be written s.r_ v':.(:+ 45dx ,tY dz (.:r,,_ T-x, ,,,5 a o -"
The principles presented in No. invariant, in the sense of No. 305
305
.r,
D
n 2 and
small
R play
the
role
which
The tern under the sign x 2 -- i.e., for points
origin,
or
very
close
to
the
_ and
enable
nt
-_
x2
us
deduce
the
following
n,
O played
f is essentially of the half-plane
to
in
the
analysis
positive, except which are very
of No.
305.
for very far from 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
place our half-plane by a simply connected curve on a surface. choose this curve on a surface in the following way.
turn
If x 2 is very small, the eccentricity is very small, in the opposite direction. The principles presented
applicable, first type ties
and we may affirm which will clearly
could We
re-
shall
and the two planets in No. 40 are
the existence of a periodic solution of the _377 satisfy the following conditions: The quanti-
¢'x_cosys,
¢x_siny_,
x,,
cosy,,
siny,
are periodic functions of the time t. These functions depend on _ and on the energy constant C. They may be developed in powers of _; the period T also depends on _ and on C. The angle Yl increases by 2_ when t increases by a period. that we have With have when
370
Finall_, _/_ cos x 2 = 0 for _ = 0.
our
called o, _ is very
method
of
Y2
and
representation,
is represented by small, this curve
V_
this
sin
y2
are
periodic
a closed curve K. is displaced very
divisible
solution,
Since little
by
_,
which
so
I
x 2 is very small from the Z axis.
It may be stated that it is displaced from it very little, in the that a circle having a very large radius is displaced very little straight line. Every point on the K curve is either very far from or very close to the Z axis. Under
this
assumption,
our
curve
on
a surface
S would
have
same way from a the origin
the
curve
K for the perimeter, and it would be displaced very little from the halfplane Y = O, X > O, 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 equations -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 from the first method. If we reflect on this a little, we will find this difficulty is similar to that in Chapter XII. We must therefore the change in variables similar to that performed in No. 145. Let
us
first
set
_,= ¢,_ y>y:--_.
yo+_>y>Yi;
N_
et
N'6;
yt>y:>Yo.
(6)
to
n is y".
on
which
Nt;
e,
our
the
chosen
and
/403
it
equations
for
NI
and N'_ coincide, doubly asymptotic
solution
(5)
for
the intersection will solution which will be
t = -%
and
very
close
to
the
t = +_.
to determine
this
intersection,
let
us
compare
the
equations
N'4
intersection
will
dSt.i,
¢ ---_-x
clearly
,
p=po'4-*-'--_,
be
given
d(S,.,-St.,) dx
Sl.l - St.4 is a function negative
y
N:;
dSH
the
depends
ct
p =/_o+
and
that than
Na et
periodic
solution
clearly
values
corresponds
assume larger
approximation:
For example, if the surfaces correspond to a heteroclinous, close
the
which
I shall of y is
E strives
following
Nt
very
of
y'. Let R n be its n IK consequent. enough that the corresponding value
increases
may
following
whole
powers
of x
and
y,
by
(9)
=o.
which
may
be
developed
in
positive
and
of _ielx.
The fact that it is a periodic function of x is important fore has at least a maximum and a minimum. Equation (9) least two solutions, which means that there are at least solutions In ponding to the
394
• the
same
to the surfaces
The
to us. It theretherefore has at two heteroclinous
way,
it
could
intersections N2 and N' 3,
preceding
analysis
be
of and does
shown
the two not
that
there
surfaces N4 corresponding yield
the
are
two
and N'2, to the homocllnous
solutions
corres-
two corresponding surfaces N3 and N'I. solutions.
404.
For
example,
let
us
Fo
set
--p--ql+2H.
sin'Y--Y°sin_Y--Y' 2
2
FI ----F"cosx sin(y --Yo) sin(y
The periodic solutions strive for t = -_ and
It will
be
noted
(5) and (6) toward t = +_ are then
that,
= 0, the function and does not depend F therefore has the
for
x=t, x =:t,
p=q=o, p=q:=o,
_ = 0,
F may
quantities
two dx _
solutions and
dy dt
which
were
these
called
the
heteroclinous
solutions
(5) --
Y:=Y0, y=y_. be
reduced
to -p
- q2.
Therefore,
for
p and q, function
and
(6) both
correspond
to
the
same
value
of
the
i.e.
quantities
n I and
/404
not be content with this example, which proves having the form considered in No. 13 can have
_Ix di ::"
However,
which
F depends only on variables of the first series on variables of the second series x and y. The form considered in Nos. 13, 125, etc.
Nevertheless, we shall that the canonical equations heteroclinous solutions. The
--Yl).
dx _,
dy dr.... dy dt
are
o. nothing
else
than
the
numbers
n 2 above.
Therefore, we find that doubly asymptotic solutions exist, which come infinitely close to two different periodic solutions for t = -_ and t = 4_. However, these two periodic solutions correspond to the same values of the numbers n I and n 2.
with
Therefore, equations
I shall formulate another example, in which we shall having the same form as those presented up to No. 13,
which have doubly asymptotic solutions coming arbitrarily ic solutions which are not only different, but correspond nl of the ratio--. n2 Unfortunately, I would like to show that values of U which are close to i, but I still fact that they also exist for small values of 405.
We
shall
take
two
pairs
of
conjugate
deal and
close to two to different
these solutions am not able to _.
periodvalues
exist for establish the
variables
395
v
or
J405 xl, y,;
by
x,, ,ys,
setting b = I/_-_l cos,y,;
This
change
equations.
We
in
variables
shall
does
_t
=
not
%/_zi
alter
pendent
shall
assume
of
and
Yl
that
Y2,
F 0 is
the
and
that
function i x2 = _, we
--0_
_
assume
that
for
x 2
=
shall
assume
It have
F
that
follows =
The
F0,
a
from
our
first
