Instability of periodic orbits in restricted and reduced three-body problems using mapping by fixed point metho

Hitzl, D. L.

White, J.

English

NASA Technical Reports Server

I 
I 
I 
1 
I 
I 
1 
I 
I 
I 
I 
I 
I 
II 
I 
I 
I 
I 
1 
LMSC/HREC A783146 
LOCKHEED MISSILES & SPACE COMPANY 
HUNTSVILLE RESEARCH & ENGINEERING CENTER 
HUNTSVILLE RESEARCH PARK 
4800 BRADFORD DRIVE, HUNTSVILLE, ALABAMA 
RESEARCH IN STABILITY 
O F  PERIODIC MOTIONS 
QUARTERLY PROGRESS 
5 May to 4 August 1966 
Contract NAS8 - 203 2 3 
Prepared f o r  George C. Marshall Space Flight Center 
by 
D. L. Hitzl and John White 
I 
x ,  j .  . 
3 .-y I APPROVED BY: yi J .  
K. L. Remmler, Supvr., 
Vehicle Dynamics 
/ I’ 
G. E. ChristopheP, Mgr., 
Dynamics &-Guidance 
/ - /I ,,&7&y&/ \ 
JJ S. Farr ior ,  Resident Manager - // Huntsville R E Center 
/’ 
https://ntrs.nasa.gov/search.jsp?R=19660029545 2020-03-16T17:25:08+00:00Z
LMSC/HREC A783146 
FOREWORD 
The principal objective of Contract NAS8-20323, Research in Stability 
of Periodic Motions, is to derive exact analytical results concerning the 
degree of instability of certain periodic orbits in the restricted and reduced 
three body problems. However, before this problem can be formulated, 
considerable background material  must be assimilated. 
summarizes the background material studied during this period of contract 
pe r f o r mance . 
This report 
This investigation is being performed by personnel of Lockheed 
Missiles & Space Company, Huntsville Research & Engineering Center 
for the Aero-Astrodynamics Laboratory of the George C. Marshall Space 
Flight Center. 
ii 
I 
ti 
i 
I 
I 
1 
1 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
1 
LMSC/HREC A783146 
SUMMARY 
During this reporting period, Chapters 20, 21, 22, 26 and 27 of 
Dr. Carl  Ludwig Siegel's Vorlesungen uber Himmelsmechanik were read. 
In Chapter 22, area-preserving transformations are discussed in view of 
application to proving Birkoff's fixed point theorem. The normal form is 
established and its physical interpretation discussed. In particular, the 
idea of invariant curves is introduced in their simplest form. The impor- 
tance of these invariant curves in studying the stability of periodic motions 
is pointed out. 
The Liapunov and Dirichlet theorems on the stability of equilibrium 
solutions of a se t  of differential equations a r e  discussed. 
iii 
LMSC/HREC A783146 
CONTENTS 
FOREWORD 
SUMMARY 
INTRODUCTION 
TECHNICAL DISCUSSION 
FUTURE WORK 
iv 
Page 
ii 
iii 
1 
2 
9 
I 
' I  
I 
LMSC/KREC A783146 
INTRODUCTION 
Two methods of establishing the existence of periodic solutions to the 
problem of three bodies has met with distinguished success. 
method, Foincarg's continuity method, was described briefly in the first 
quarterly progress report. 
to Poincarg) has been a major topic for study during the succeeding quarter, 
reported herein. The establishing of the fixed point method as a useful tool 
is largely due to Birkhoff. The basic concept behind this method is that the 
consecutive intersections of the solution with, in general, a hyperplane in 
phase space constitutes a mapping of that hyperplane into itself. Periodic 
solutions, after one or  more revolutions i f  necessary, return to intersect the 
hyperplane in the same point. 
fixed points in the mapping, and also, obviously, the converse is true. 
task then reduces to defining a mapping and then finding or proving the exist- 
ence of fixed points. 
solution, which is not an equilibrium solution, is required; and the mapping 
is of one neighborhood onto another neighborhood of this fixed point. 
to the existence of periodic solutions is that the mapping is area-preserving, 
that is, the a reas  of the neighborhood of the fixed point and its image neigh- 
borhood are equal. 
The first 
The second, the fixed point method (also credited 
These periodic solutions a re  represented by 
The 
Jn both Poincare's and Birkhoff's work a known periodic 
The key 
During the later par t  of the reporting period the stability of equilibrium- 
solutions was studied, in particular the theorems of Liapunov and Dirichlet. 
Liapunov gives a sufficient condition for instability and a necessary condition 
for stability for the equilibrium solution of a set  of first order linear differ- 
ential equations. Dirichlet, on the other hand, gives a sufficient condition 
for stability provided that a time-dependent integral of the equations exists. 
~~~ 
LMsC/HREC A733146 
I 
TECHNICAL DISCUSSION 
The solutions 
of the differential equations 
which satisfy the conditions of Cauchy’s existence theorem, define, at  a time 
t, a mapping of the initial conditions, tk = \(to, t l ,  . . . , 4 
m 
onto xk = \(t, el, . . . , 5,). A suitably chosen domain U of m-volume > 0, 
but finite, is mapped onto a domain U: 
functional de terminant 
) (k = 1, . . , m), 
The m-volume is preserved if the 
Since at  time t = 0 the matrix (% ) = I, the unit matrix the condition (3) 
becomes % 
A = o .  
By some careful manipulations this reduces to 
This condition is obviously satisfied for a Hamiltonian system 
2 
LMSC/HREC A783146 
since the condition becomes 
n 
k= c 1 (Eyk”k-ExkY$ = O 
Now with the eventual application to  motion near a non-equilibrium periodic 
solution, we consider a solution with initial conditions tl, . . , 
trn = const., at time t which after a time t = (again intersects the hyper- 
plane 6- = const., at (1‘ . .. , tk-l,L. From continuity considerations, 
a neighborhood, B, of t l ,  . . , {m-l, tm .F contained in the hyperplane & = const. 
is mapped onto a neighborhood, B: of t;, . . . , 
of all  points of the solution connecting B to B‘as the domain U of initial condi- 
tions at time t = t After  a time t‘,Umaps onto U, Le., the s t ream tube U is 
0. 
moved on, following the trajectories, so that volume is preserved. This 
situation is depicted, in three dimensions, in Figure 1. 
8 
4m-10 t m ~  , 8 
* 0: 
m 
.(r 
rtr . Now consider the set S&-l* tL 
I 
Stream Tube 
of T r a  je ctor ie s 
f rom B to B 
Figure 1 - Diagram of Mapping Defined by the Solutions 
of a Set of Differential Equations 
3 
= const. 
LMSC/HREC A783146 
Let the points of the surface B-B and B'-B; and R and R'be the 
m-volumes of the s t ream tubes between B and B1, and B'and B; respectively. 
1 
Then since the mapping is m-volume preserving 
U = U'= U-R+R' 
... R = R' . 
In general the (m-1) volume of B # (m-1) volume of B: but we can establish 
that 
The time t has been eliminated by letting t'-0. 
dimension of the set of initial conditions even further it is assumed that a 
In order to reduce the 
time independent integral #(x) of the system (2) exists and that @ X f 0. 
m- 1 Under this assumption the invariant integral (6) becomes 
where F and F are  the appropriate subsets of B and B 1 1 respectively. 
For  the Hamiltonian system (4) E itself satisfies the requirements for 9 
and so 
on a suitable ordering of the variables, and so 
1 dv = dv 
F F. 
1 
i.e., the mapping preserves (m-2) volumes. 
4 
LMSC/HREC A783146 
In the case of the restricted problem of three bodies, this means that 
the mapping near a periodic solution preserves area. 
To take full advantage of area-preserving mappings, we must choose 
our coordinate system so that  the mapping takes on some recognizable form. 
Siege1 accomplishes this task by a coordinate transformation written as an 
infinite power ser ies  which may or  may not converge. 
mapping itself is also written as an infinite power eeriee, which, to be of use, 
must converge. 
The area-preserving 
We let our area-preserving mapping be 
x1 = a x + b y t p ( x , y )  
Y 1  = c x +  d y  + s (x ,y )*  
where p (x, y) and q(x, y) s tar t  with terms of a t  least degree two, and the origin 
(0, 0) is the fixed point of the mapping. 
the hyperbolic, parabolic and elliptic according to whether (a-d) 
Three cases a re  recognized, namely 
- 4 bc is >, 2 
=, -c 0 respectively. The matrix of coefficients e 1) of the linear terms is 
reduced to diagonal form 6":) by a linear transformation. The normal form 
in the hyperbolic case, is 
cn where 
r l  = r cosw - s sinw, s1 = r sinw t s cosw 
where 
cb 
w = c yk( r2  t e21k . 
k= 0 
I 
I 
5 
LMSC/HREC A783146 
, I  I 
‘ .I 
I 
1 
I 
where #,# a re  formal power ser ies  in the stated variables. 
We now consider the interpretation of these forms. In the hyperbolic 
case 
41 ‘11 = ttl 
i.e., if ( , q  a r e  interpreted as rectangular coordinates of t le point P # ( o ,  o), 
then P is mapped onto P1 which lies on the rectangular hyperbola passing 
through P. Further we may find some disc S = x + y 5 p ( p  > 0) in the 
domain of convergence of ( 9 )  such that all the image points of Pes fo r  
successive applications of the transformation (9) wil l  not lie within the disc 
for any point P # (0,O). 
a s  rectangular coordinates, the equations 
2 2 2  
Now in the elliptic case, again r, s are  interpreted 
= r cosw - s sinw, s = r sinw + S  cosw 
r l  1 
means that the radius vector to the point P = (r, s )  is of invariant magnitude 
and is rotated through an angle 8 dependent upon p = (r t s ) . We see,  
then, that every circle of radius p, p sufficiently small, is invariant if 4 and 
Itare convergent. 
the invariant circles and the points on the trajectories, compare to Figure 1 , B=B: 
define a closed surface and so motion is restricted to within o r  without this 
surface. 
from our elementary concepts of stability, a motion contained within a finite 
invariant curve is stable. Since in the elliptic case, convergent 4, I) implies 
all  circles of sufficiently small radius a re  invariant we  a re  not surprised to 
learn that, in the elliptic case divergence of 4,@, is, infact, general. However, 
this divergence of &t+b does not preclude invariant curves, as shown by Moser, 
nor stability. Applying our intuitive concept of stability to the hyperbolic case, 
we concluded that the hyperbolic case wil l  give instability since we can always 
find a circle such that the image of P lies outside this circle. 
2 2 1/2 
Now from continuity of the solutions (1) we conclude that 
A motion initially within th is  surface is therefore bounded, and, 
In the final paragraphs, a convergent 4: @’are generated, which reduces 
the area-preserving transformation to a form which is identical to the normal 
form to an arbi t rary power in S,Q plus a remainder t e r m  which, of course, 
is convergent if the original area-preserving transformation is convergent. 
6 
i 
8 
II 
I 
I 
8 
1 
1 
I 
I 
I 
I 
I 
I 
8 
I 
I 
I 
1 
I 
LMSC/HREC A783146 
The form considered is 
where P, Q a re  infinite ser ies  in6.q commencing with terms of degrees 
2P’ + 2. 
which states: 
This is the starting point fo r  the proof of the Birkhoff fixed point theorem 
k There exist fixed points Z # 0 of Sn such that S ZCG (k = 0, . . . , n-1) 
exist, in any neighborhood G, whichmay be as small as we please, of the 
origin in the (x, y)-plane, and for all sufficiently large integers n > no (G). 
This theorem is proven by showing that for n sufficiently large, each 
point of a continuous curve surrounding a fixed point, Z = 0, is mapped 
along the radius vector from the origin. 
so that the image curve cannot lie entirely inside nor entirely outside the 
selected curve; therefore, it has at least two fixed points. (See Figure 2. ) 
The mapping is area-preserving, 
tq 
Fixed 
Point 
\ 
I 
Figure 2 - Diagram of Radial Mapping under a n  Area-Preserving Mapping 
7 
LMSC/HREC A783146 
Stability of the equilibrium solutions of a system (1) is the topic of the 
Liapunov and Dirichlet theorems. 
by these theorems provided these solutions a re  in fact, equilibrium solutions 
in some suitably chosen coordinate system, for example, the libration points 
of the restricted problem of three bodies in rotating coordinates. 
Periodic solutions may be investigated 
If the eigenvalues of the matrixA of the linear terms o f f  (x), .. . , f  (x) 1 m 
then the Liapunov Theorem teiis us that the m' are denoted by A 1s * *  ,A 
equilibrium solution is unstable if the rea l  parts of A 1, . . . ,A, are all nonzero 
Am and that if the equilibrium solution is stable, then the rea l  parts of A,, . . . , 
a r e  a l l  zero. 
Dirichlet gives a sufficient condition for the stability of the equilibrium 
solution of the system (1) provided it possesses a time independent integral 
g(x). This theorem states that: 
If the system (1) possesses a time independent integral g(x) which has 
a relative extremum in the strict  sense at x = 0, then the equilibrium solution 
x = 0 is stable. 
8 
~~~ ~ ~ 
LMSC/HREC A783146 
FUTURE WORK 
Planned for the future is a continued study of the normalization of the 
Hamiltonian and existence of invariant curves under area-preserving mappings 
with particular emphasis on applications to the stability of periodic orbits. This 
will include portions of Siege1 as well as  appropriate publications by Dr. Moser 
and Dr. Arenstorf. 
9 


Research in stability of periodic motions Quarterly progress report, 5 May - 4 Aug. 1966

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server