Table Of Contentde Gruyter Expositions in Mathematics 17
Editors
O.H.Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, The University of Melbourne, Parkville
R. O. Wells, Jr., Rice University, Houston
de Gruyter Expositions in Mathematics
1 The Analytical and Topological Theory of Semigroups, K. H. Hofmann,
J. D. Lawson, J. S. Pym (Eds.)
2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues
3 The Stefan Problem, A.M.Meirmanov
4 Finite Soluble Groups, K. Doerk, T. O. Hawkes
5 The Riemann Zeta-Function, A.A.Karatsuba, S.M. Voronin
6 Contact Geometry and Linear Differential Equations, V. R. Nazaikinskii,
V. E. Shatalov, B. Yu. Sternin
1 Infinite Dimensional Lie Superalgebras, Yu.A.Bahturin, A.A.Mikhalev,
V. M. Petrogradsky, M. V. Zaicev
8 Nilpotent Groups and their Automorphisms, E. I. Khukhro
9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug
10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini
11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon,
G. Zhao
12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and
Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub
13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A. Na-
zarov, B. A. Plamenevsky
14 Subgroup Lattices of Groups, R. Schmidt
15 Orthogonal Decompositions and Integral Lattices, A.I.Kostrikin,
P. H. Tiep
16 The Adjunction Theory of Complex Projective Varieties, M. Beltrametti,
A. Sommese
The Restricted 3-Body Problem:
Plane Periodic Orbits
by
Alexander D. Bruno
With a preface by Victor G. Szebehely
Translated from the Russian by Balint Erdi
w
DE
G
Walter de Gruyter · Berlin · New York 1994
Author Translator
Alexander D. Bruno Balint Erdi
Institute of Applied Mathematics Department of Astronomy
Miusskaja sq. 4 Loränd Eötvös University
Moscow 125047, Russia H-1083 Budapest, Hungary
7997 Mathematics Subject Classification: 70-02; 70F07, 34C25, 58F22, 58F40, 70F15,
70H05
Key words: Hamiltonian systems, normal forms, periodic solutions, perturbation bifur-
cations, collisions, resonance
Title of the Russian original edition: Ogranichennaya zadacha trekh tel: Ploskie periodi-
cheskie orbity. Publisher: Nauka, Moscow 1990
With 93 figures and 12 tables
© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Brjuno, Aleksandr Dmitrievich.
[Ogranichennaja zadacha trekh tel. English]
The restricted 3-body problem : plane periodic orbits / by
Aleksandr D. Bryuno ; translated from the Russian by Balint Erdi.
p. cm. — (De Gruyter expositions in mathematics ; 17)
Includes bibliographical references and index.
ISBN 3-11-013703-8
1. Hamiltonian systems. 2. Three-body problem. I. Title.
II. Title: Restricted three-body problem. III. Series.
QA614.83.B7513 1994
514'.74-dc20 94-27982
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Brjuno, Aleksandr D.:
The restricted 3 body problem: Plane periodic orbits / by
Alexander D. Bruno. With a pref. by Victor G. Szebehely. Transl.
from the Russ. by Balint Erdi. - Berlin; New York : de Gruyter, 1994
(De Gruyter expositions in mathematics ; 17)
ISBN 3-11-013703-8
NE: Ogranicennaja zadaca trech tel: Ploskie periodiceskie orbity
<engl.>;GT
© Copyright 1994 by Walter de Gruyter & Co., D-10785 Berlin.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced in any form or by any means, electronic or mechanical, including photocopy,
recording, or any information storage and retrieval system, without permission in writing from
the publisher.
Printed in Germany.
Disk Conversion: Lewis & Leins, Berlin. Printing: Gerike GmbH, Berlin.
Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.
Preface by Victor Szebehely
"Take time" said Albert Einstein referring to this as one of the important aspects
of scientific creation. Indeed, it took us 200 years from Newton's Principia to
Poincare's theory of non-integrability of the problem of three bodies.
In my book Theory of Orbits, I considered omitting Chapter 8 dedicated to
periodic orbits because I felt that this subject deserves a volume rather than a
chapter, but "I kept it to serve as an outline of another book devoted to periodic
orbits." (cf. Theory of Orbits, p. 384). So now Professor Bruno enters the picture
and 27 years after my book was published presents us with a wonderful, clear and
precise book on periodic orbits in the restricted problem of three bodies.
It is difficult to argue against the idea that periodic orbits are of the utmost
importance in non-integrable dynamical systems. Our modern-day approach is
to turn often to our computers when "unsolvable" problems are faced. (I prefer
asking for the help of my graduate students and assistants who are less inhibited
and more competent using computers than analytic approaches.)
So what happens when high precision, long-time orbits are to be obtained for
non-integrable dynamical systems? There seem to be three possibilities for the
outcome:
(i) we accept the results of numerical integrations without applying controls
such as reversed integration and constancy of integrals; indeed if such checks
are applied the error accumulations would indicate that our results are mean-
ingless;
(ii) we admit that the problem cannot be solved and no precise long-time orbits
can be established, or
(iii) we restrict our work to finding periodic orbits as the only solutions valid for
arbitrary long time.
In other words, the only solutions valid for arbitrary long time of non-integrable
dynamical systems are periodic orbits.
We should not be surprised to find that periodic orbits received the attention of
our greatest scientists concerned with analytical mechanics and dynamics. And I
do not mean only the establishment of actual periodic orbits but the detailed study
of the analytical and geometrical properties of such orbits.
Professor Bruno selected the best subject in dynamics: periodic orbits. He em-
phasizes the most important dynamical system from the point of view of celestial
vi Preface by Victor Szebehely
mechanics, the restricted problem of three bodies. The problem is essential to
prepare trajectories for space dynamics applications, for the study of the behavior
of small and large members of the solar system, and indeed, to establish existence
criteria for planets in binary stellar systems. This last problem brings cosmology
into the picture because of the high percentage of binary stars in our galaxy. Con-
sequently, to find other planetary systems requires the study of periodic orbits in
the restricted problem of three bodies.
Professor Bruno's background as a mathematician will be clear to the reader.
Our fields of celestial mechanics and analytical dynamics attracted many math-
ematicians (Arenstorf, Arnol'd, Birkhoff, Levi-Civita, Moser, Poincare, Sinai,
Stiefel, Wintner, to mention a few). Professor Bruno joins this group and writes
with basic understanding and many explanatory remarks, instead of hiding behind
undefined mathematical terms.
The Notes following every Chapter offer some interesting and important his-
torical background and show the deep understanding and familiarity of the author
with the subject of this book.
In conclusion, we should be grateful to Professor Bruno for joining the family
of scientists who dared to enter the mine-field (i.e. singularities) of celestial me-
chanics. Indeed, Professor Bruno now becomes a descendant of our past heroes:
Copernicus, Newton, Euler, Lagrange, Laplace, Gauss, Hamilton, Hill, Tisserand,
Hagihara, Brouwer and Kolmogorov, just to add a few names to our previous list.
Those of us who wish to establish real and exact solutions to this important
problem and are, consequently, dedicated to find periodic orbits in the three-
body problem are most grateful for Professor Bruno's beautiful and important
contribution and to Dr. Erdi's excellent and valuable translation.
Austin, Texas, January 1994 Victor Szebehely
Table of Contents
Preface v
Basic notation xi
Introduction 1
Parti
Mathematical means n
Chapter I
The normal form of a Hamiltonian system 13
1 The normal form of a linear system 14
2 Nonlinear normalization 25
3 Lowering the number of degrees of freedom 35
4 Periodic and conditionally-periodic solutions 40
5 Generalizations 44
6 Notes 48
Chapter II
Normalization of a Hamiltonian system near a cycle or a torus 54
1 The normal form of a linear Hamiltonian system with periodic coefficients 54
2 The normal form in a neighbourhood of a periodic solution 62
3 The neighbourhood of an invariant torus 75
4 A system with two degrees of freedom 80
5 Notes 87
viii Table of Contents
Part II
Solutions of the limiting problem (μ = 0) 93
Chapter III
Periodic solutions and arc solutions 95
1 Formulation of the restricted three-body problem 95
2 The two-body problem 97
3 The restricted three-body problem for μ = 0 113
4 Notes 132
Chapter IV
Properties of arc solutions of the families S and T 133
N
1 Orbits of the families S 133
2 The structure of the set of characteristics of the families 5 and TN 152
3 Values of the Jacobian constant 168
4 Notes 176
Chapter V
Extrema of the Hamiltonian function on families of arc solutions 178
1 The function G 178
2 Partial derivatives of the function G 182
3 Classification of the segments of the characteristics 186
4 Estimation of the number of intersections of a segment of
a characteristic with a level line of the function G 187
5 The domain p 190
3
6 Conclusion 193
Chapter VI
Trajectories of collisions 194
1 The problem of two bodies P\ and P 195
3
2 Trajectories of collision with P 206
2
3 Arc solutions with consecutive collisions 222
4 Isolation of critical solutions 228
5 Notes 230
Table of Contents ix
Part ψ
Regular generating solutions 233
Chapter VII
Solutions of the second kind 235
1 Sets of periodic solutions 235
2 The restricted three-body problem 242
3 Averaging of the perturbing function 245
4 Computation of generating families 249
5 Concerning the programs 253
6 Other properties of the families E 254
N
7 Limits of generating families 255
8 Conditionally-periodic solutions of the second kind 262
9 Notes 263
Chapter VIII
Solutions of the first kind 266
1 Reduction to the normal form 266
2 Integration of the normal form for rational λ = p/q 275
3 Application to the restricted problem 283
4 Conclusion 288
5 The neighbourhood of stationary points 290
6 Notes 295
Chapter IX
Generating arc solutions from C\C\ 298
1 Generating solutions of collisions with P\ 298
2 Computation of families of generating arc solutions 305
3 Asymptotic s of families of generating arc solutions 310
4 Concerning periodic solutions with collisions 314
5 Notes 315
Table of Contents
Chapter X
Perturbations of hyperbolic orbits 317
1 The normal form 317
2 Application to the restricted problem 321
3 Notes 325
Appendix
Tables of generating families and their perturbations 327
Bibliography 347
Author index 357
Subject index 359
