Periodic Orbits, Stability and Resonances: Proceedings of a Symposium Conducted by the University of São Paulo, the Technical Institute of Aeronautics of São José Dos Campos, and the National Observatory of Rio De Janeiro, at the University of São Paulo,

539 Pages·1970·38.44 MB·English

by  André Deprit

PERIODIC ORBITS, STABILITY AND RESONANCES
PERIODIC ORBITS,
STABILITY AND RESONANCES
PROCEEDINGS OF A SYMPOSIUM CONDUCTED BY THE UNIVERSITY OF
SAO PAULO, THE TECHNICAL INSTITUTE OF AERONAUTICS OF
SAO JOSE DOS CAMPOS,AND THE NATIONAL OBSERVATORY OF
RIO DE JANEIRO, AT THE UNIVERSITY OF SAO PAULO,
SAO PAULO,BRASIL,4-12 SEPTEMBER, 1969
Edited by
G. E. O. GIACAGLIA
Escola Politecnica, University of Sao Paulo and
Instituto de Matematica, University of Campinas,
Sao Paulo, Brasil
D. REIDEL PUBLISHING COMPANY
DORDRECHT-HOLLAND
Library of Congress Catalog Card Number 74-124848
SBN Number 9027701709
lSBN-13: 978-94-010-3325-1 e-lSBN-13: 978-94-010-3323-7
DOl: 10.1007/ 978-94-010-3323-7
All Rights Reserved
Copyright © 1970 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1st edition 1970
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
"There once was a comet quite old
Whose core grew gradually cold
It began to librate
With amplitude great
And now has a shell of green mould."
W. T. KYNER
PREFACE
The subjects of resonance and stability are closely related to the problem of evolution
of the solar system. It is a physically involving problem and the methods available to
mathematics today seem unsatisfactory to produce pure non linear ways of attack.
The linearization process in both subjects is clearly of doubtful significance, so that,
even if very restrictive, numerical solutions are still the best and more valuable sources
of informations. It is quite possible that we know now very little more of the entire
problem that was known to Poincare, with the advantage that we can now compute
much faster and with much more precision.
We feel that the papers collected in this Symposium have contributed a step forward
to the comprehension of Resonance, Periodic Orbits and Stability. In a field like this,
it would be a surprise if one had gone a long way toward that comprehension, during
the short time of two weeks. But we are sure that the joint efforts of all the scientists
involved has produced and will produce a measurable acceleration in the process.
If this is true it will be a great satisfaction to us that this has happened in Brasil. The
Southern Hemisphere in America has now begun to participate actively in the Astro
nomical Society and for this, we are grateful to everyone who has helped.
G. E. O. GIACAGLIA
December, 1969
FOREWORD
This volume presents the invited and contributed papers of the International Sympo
sium on 'Periodic Orbits, Stability and Resonances' held in Sao Paulo from the 4th
to the 12th of September, 1969. The papers are arranged in the same order as they
were presented.
The purpose of the Symposium was to congregate scientists from all over the world
to discuss one of the most exciting topics in Celestial Mechanics. It served to provide
stimulation for researches in Astronomy in the Southern Hemisphere. In the overall,
it resulted also as a catalyzer for development of new methods in Applied Mathematics.
We extend our thanks to the following Institutions who have financially supported
the expenses for the participants of the Symposium:
a
Funda'Yao de Amparo Pesquisa do Estado de Sao Paulo
Conselho Nacional de Pesquisas
Prefeitura do Municipio de Sao Paulo
Horsa - Hoteis Reunidos S.A.
Varig - Via'Yao Aerea Riograndense
Constru'Yoes e Comercio Camargo Correa S.A.
Ceramica Sanitaria Porcelite S.A.
Ford Motors do Brasil S.A.
Chrysler do Brasil S.A.
We have deeply appreciated the prompt collaboration of the invited lecturers and
the scientific support received by the International Astronomical Union. It is our
pleasure to include in this volume a letter by Prof. C. de Jager, Assistant Secretary
ofthe IAU, which was read in the opening session of the Symposium.
The Executive Committee of the Symposium was composed by
Prof. AbraMo de Moraes, President
Prof. Luiz Muniz Barreto
Prof. Sylvio Ferraz Mello
Prof. Victor Szebehely, Coordinator in U.S.A.
Prof. G. E. O. Giacaglia, General Coordinator
Dr. Iussef Hana Abduch, General Secretary
It is our duty to acknowledge the assistance of Mrs. M. Olympia A. F. Fran'Ya who
arranged the social activities, Mrs. Mary Y ohanna who arranged the reception scheme
and Mr. G. Schmidt, Assistant Secretary of the Symposium.
LETTER OF PROFESSOR DE JAGER
TO THE ORGANIZING COMMITTEE
At the occasion of the Symposium on 'Periodic Orbits, Stability and Resonances',
to be held in Sao Paulo, from 4 till 12 September 1969, the International Astronomical
Union wishes to congratulate you and the other Members of the Executive Com
mittee of this Symposium for the interesting meeting that you have organized. We
send you our best wishes for a successful and highly stimulating scientific meeting.
The International Astronomical Union regrets that too few large astronomical
meetings take place in Latin America and very much welcomes the present Symposium.
The Union expresses the hope that this may stimulate young scientists in Brasil and
other Latin-American countries in their study of astronomy, and hopes that the
Symposium may contribute to the further development of astronomy in your continent.
For the Executive Committee of the
International Astronomical Union,
c. DE JAGER,
Assistant General Secretary
TABLE OF CONTENTS
Preface VII
Foreword IX
Letter of Professor De Jager to the Organizing Committee XI
ANDRE DEPRIT and JACQUES HENRARD / The Trojan Manifold-Survey and
Conjectures 1
P. J. MESSAGE / On Asymmetric Periodic Solutions of the Plane Restricted
Problem of Three Bodies 19
EUGENE RABE / Two New Classes of Periodic Trojan Librations in the Elliptic
Restricted Problem and Their Stabilities 33
J. SCHUBART / Minor Planets on Commensurable Orbits with Approaches to
Jupiter 45
HARRY POLLARD / Disintegration and Escape 53
O. GODAR T / Secular Variations Determined by a Surface of Section 56
DONALD G. SAARI/On Bounded Solutions of the n-Body Problem 76
AHMED A. KAMEL and JOHN V. BREAKWELL / Stability of Motion near Sun-
Perturbed Earth-Moon Triangular Libration Points 82
G. E. O. GIACAGLIA and L. N. F. FRANC;A / Motion near Sun Perturbed Earth-
Moon Collinear Equilibrium Points (Status Report) 91
G. E. O. GIACAGLIA and PAUL E. NACOZY / Resonances in the Elliptic
Restricted Problem 96
B. E. SCHUTZ and B. D. TAPLEY / Numerical Studies of Solar Influenced
Particle Motion near the Triangular Earth-Moon Libration Points 128
w. J. KLEPCZYNSKI/ On Using Minor Planets Close to the 2: 1, 3:2, 4:3
Commensurabilities to Determine the Mass of Jupiter 143
B. G. MARSDEN / On the Relationship Between Comets and Minor Planets 151
s. FERRAZ-MELLO / Absolute Orbits and Jupiter's Great Satellites (Progress
Report) 164
MARKO SVEC / Existence of Periodic Solutions of Differential Equations of
Second Order 168
G. BOZIS / Sets of Collision Periodic Orbits in the Restricted Problem 176
LLOYD CARPENTER / Periodic Orbits in Trigonometric Series 192
SAMUEL PINES / Bifurcation Limits for the Existence of Periodic Orbits 210
PEDRO E. ZADUNAISKY / On the Accuracy in the Numerical Computation of
Orbits 216
XIV TABLE OF CONTENTS
RICHARD B. BARRAR / On the Non-Existence of Transformations to Normal
Form in Celestial Mechanics 228
WILLIAM A. MERSMAN / A Unified Treatment of Lunar Theory and Artificial
Satellite Theory 232
JOHN P. VINTI / Stability of Free Rotation of a Rigid Body 259
J. M. A. DANBY / Wild Dynamical Systems, and the Role of Two or More Small
Divisors 272
J. KEVORKIAN / The Planar Motion of a Trojan Asteroid 286
JURIS VAGNERS / On the Long-Term Evolution of Lunar Satellite Orbits 304
J. L. SERSIC / Transient Annular Structures in Exploding Galaxies 314
G. CONTOPOULOS / Resonance Phenomena in Spiral Galaxies 322
G. COLOMBO and F. A. FRANKLIN / On the Evolution of the Solar System and
the Pluto-Neptune Case 328
ROBERT EASTON / Flows near Isolated Invariant Sets in Dimension 3 332
w. M. OLIV A / Dynamical Systems on Manifolds 337
NELSON ONUCHIC / On a Criterion of Instability for Differential Equations
with Time Delay 339
DAVID FISHER / The Libration Case ofthe Stellar Problem of Three Bodies 343
MICHEL HENON and MONIQUE GUYOT / Stability of Periodic Orbits in the
Restricted Problem 349
E. MYLES STANDISH, JR. / New Periodic Orbits in the General Problem of Three
Bodies 375
VICTOR SZEBEHEL Y / New Families of Periodic Orbits in the General Planar
Problem of Three Bodies 382
WILLIAM H. JEFFERYS / Stability and Resonances in the Restricted Problem 397
v. A. BRUMBERG / Application of Hill's Lunar Method in General Planetary
Theory 410
Y. KOZAI/ Stationary and Periodic Solutions for the Restricted Problem of
Three Bodies in Three-Dimensional Space 451
C. FREDERICK PETERS / Motion ofa Space Probe near an Oblate Planet 469
BORIS GARFINKEL / On the Ideal Resonance Problem 474
PETER HAGEDORN / Parametric Resonance in Certain Nonlinear Systems 482
GEN-ICHIRO HORI / Resonances in Duffing's Problem 493
w. T. KYNER / Passage Through Resonance 501
G. E. o. GIACAGLIA / Two Centers of Libration 515
THE TROJAN MANIFOLD - SURVEY AND CONJECTURES
ANDRE DEPRIT and JACQUES HENRARD
Boeing Scientific Research Laboratories. Seattle, Wash., U.S.A.
Abstract. Recent results concerning the families of periodic orbits emanating from the triangular
equilibrium L4 are interpreted in an attempt to establish the evolution of these manifolds as the mass
ratio varies from Routh's critical value down to its value in the system sun-jupiter.
We show in this report how recent results concerning the equilateral equilibrium L4 in the planar
restricted problem of three bodies point to some conjectures about the genealogy of families of
periodic orbits, and especially about their evolution through a resonance.
The new results discussed here are related, in the local, to Liapunov's theorem about the emergence
of a family of periodic orbits from an eqUilibrium, and, in the global, to conjectures about the natural
termination of the family of Trojan orbits. These results clarify in part the role ofresonances.
The state of the problem prior to the year 1966 has been covered by Professor Szebehely in his
Theory 0/ Periodic Orbits. The present report restricts itself to contributions made in the last three
years.
1. Introduction
Jl. is the mass ratio so dimensioned that the interval 0 < Jl. ::::;; -1 covers the range of values
considered in the restricted problem from the degenerate case Jl.=0 which is the planar
problem of two bodies to the symmetric case Jl. = -1 which is the Copenhagen case.
C is the Jacobian constant defined so that, whatever the mass ratio Jl. may be, it is
equal to 3 at the equilibrium L4•
For mass ratios in the interval
I: 0 < Jl. < Jl.I = 0.038 520897 ...
the flow of trajectories in the vicinity of L4 is described by a conservative irreversible
Hamiltonian function with two degrees of freedom
:Yl' == :Yl'(I, s, L, S;Jl.) =:Yl'2 + L :Yl'n (1)
where n~3
and, for any n ~ 3, :Yl'n is a homogeneous polynomial of degree n in LI/2 and SI/2, its
coefficients being finite trigonometric sums in the arguments I and s with the d' Alem
bert characteristic. The frequencies being such that
o < A. < 1//'1 < < 1,
(J
there exists a sequence of critical mass ratios
(k ~ 1),
such that
Let Kbe the subset of I cleared of the critical mass ratios.
G. E. O. Giacaglia (ed.). Periodic Orbits. Stability and Resonances. 1-18. All Rights Reserved.
Copyright © 1970 by D. Reidel Publishing Company. Dordrecht-Holland