'

CONTEMPORARY MATHEMATICS

198

Hamiltonian Dynamics and Celestial Mechanics

A Joint Summer Research Conference on Hamiltonian Dynamics and Celestial Mechanics

June 25--29 I 1995 Seattle I Washington

Donald G. Saari Zhihong Xia

Editors

Selected Titles in This Series

198 Donald G. Saari and Zhihong Xia, Editors, Hamiltonian dynamics and celestial mechanics, 1996

197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry, 1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms

of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, Lie algebras and their

representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and related

topics, 1996 192 Tomek Bartoszynski and Marion Scheepers, Editors, Set theory, 1995 191 Thong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally, Jr., Editors,

Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F. Ghaleb, Editors,

Mathematical analysis, wavelets, and signal processing, 1995 189 S. A.M. Marcantognini, G. A. Mendoza, M.D. Moran, A. Octavio, and W. 0. Urbina, Editors,

Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors, Homotopy theory

and its applications, 1995 187 G. W. Brumfiel and H. M. Hilden, S£(2) representations of finitely presented groups,

1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka Ihara, and Helmut

Voelklein, Editors, Recent developments in the inverse Galois problem, 1995 185 Raul E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberta Salinas, Editors,

Multivariable operator theory, 1995 184 L. A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International

Conference on Algebra, 1995 183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications

of hypergroups and related measure algebras, 1995 182 Selman Akbulut, Editor, Real algebraic geometry and topology, 1995 181 Mila Cenkl and Haynes Miller, Editors, The Cech Centennial, 1995 180 David E. Keyes and Jinchao Xu, Editors, Domain decomposition methods in scientific

and engineering computing, 1994 179 Yoshiaki Maeda, Hideki Omoro, and Alan Weinstein, Editors, Symplectic geometry and

quantization, 1994 178 Helime Barcelo and Gil Kalai, Editors, Jerusalem Combinatorics '93, 1994 177 Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr., Editors,

Representation theory and analysis on homogeneous spaces, 1994 176 David Ballard, Foundational aspects of "non"standard mathematics, 1994 175 Paul J. Sally, Jr., Moshe Flato, James Lepowsky, Nicolai Reshetikhin, and Gregg J.

Zuckerman, Editors, Mathematical aspects of conformal and topological field theories and quantum groups, 1994

174 Nancy Childress and Jolm W. Jones, Editors, Arithmetic geometry, 1994 173 Robert Brooks, Carolyn Gordon, and Peter Perry, Editors, Geometry of the spectrum,

1994 172 Peter E. Kloeden and Kenneth J. Palmer, Editors, Chaotic numerics, 1994 171 Riidiger Giibel, Paul Hill, and Wolfgang Liebert, Editors, Abelian group theory and

related topics, 1994

http://dx.doi.org/10.1090/conm/198

CoNTEMPORARY MATHEMATICS

198

Hamiltonian Dynamics and Celestial Mechanics

A Joint Summer Research Conference on Hamiltonian Dynamics and Celestial Mechanics

June 25-29, 1995 Seattle, Washington

Donald G. Saari Zhihong Xia

Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, managing editor Andy Magid

Clark Robinson Michael Vogelius Peter M. Winkler

This volume contains the refereed proceedings of the 1995 Joint Summer Re-search Conference on Hamiltonian Dynamics and Celestial Mechanics held in Seat-tle, Washington, from June 25 to 29, 1995. The conference was sponsored by the National Science Foundation, grant DMS-9221892-003. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

1991 Mathematics Subject Classification. Primary 70Fxx, 70Hxx, 58Fxx.

Library of Congress Cataloging-in-Publication Data Hamiltonian dynamics and celestial mechanics / Donald G. Saari, Zhihong Xia, editors.

p. em. -(Contemporary mathematics; 198) Papers from the AMS-IMS-SIAM Joint Summer Research Conference on Hamiltonian Dynam-

ics and Celestial Mechanics, held June 1995 at the University of Washington, Seattle. Includes bibliographical references. · ISBN 0-8218-0566-5 (alk. paper) 1. Hamiltonian systems-Congresses. 2. Celestial mechanics-Mathematics-Congresses.

I. Saari, D. (Donald) II. Xia, Zhihong. III. AMS-IMS-SIAM Joint Summer Research Con-ference on Hamiltonian Dynamics and Celestial Mechanics (1995 : University of Washington) IV. Series: Contemporary mathematics (American Mathematical Society); v. 198. QA614.83.H337 1996 514'.74-dc20 96-9186

CIP

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission~ams.org.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

© 1996 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

10987654321 01 00 99 98 97 96

Contents

Preface vii

Qualitative Study of the Parabolic Collision Restricted Three-body Problem JOSEP M. CORS AND JAUME LLIBRE 1

Singularities in the Newtonian n-body Problem DONALD G. SAARI AND ZHIHONG XIA 21

A Variational Approach to Multibump Solutions of Differential Equations PAUL H. RABINOWITZ 31

Melnikov Method for Autonomous Hamiltonians CLARK ROBINSON 45

Exponentially Small Transversality in the Rapidly Forced Pendulum NYAYAPATI PREM KUMAR SWAMI 55

The Collinear One-bumper Two-body Problem SAMUEL R. KAPLAN 87

Limiting Mass Distributions of Minimal Potential Central. Configurations PETER W. LINDSTROM 109

The Symmetric Central Configurations of Four Equal Masses ALAIN ALBOUY 131

Central Configurations in the Charged Three Body Problem ERNESTO PEREZ-CHAVELA, DONALD G. SAARI, ANTONI SUSIN, AND

ZHIMING YAN 137

Reduction in the Rotating Kepler Problem and Related Topics MARTIN KUMMER 155

Asymptotic Behavior of Escape Solutions of Mechanical Systems with Polynomial Potentials

MANUEL FALCONI AND ERNESTO A. LACOMBA 181

More on the Heteroclinic Orbits for the Monotone Twist Maps QIUDONG WANG 197

v

vi CONTENTS

Transition Maps of Homoclinic Orbits and Resonances near Bifurcations of Circle Maps

TODD R. YOUNG 207

Resonance Transition and Instabilities; A Numerical Study of the Restricted Three-body Problem

XINHAO LIAO, DONALD G. SAARI, AND ZHIHONG XIA 219

Directions of Hamiltonian Dynamics and Celestial Mechanics SAMUEL R. KAPLAN, RICHARD CusHMAN, SEN Hu, JAUME LLIBRE,

CHRISTOPHER McCORD, DONALD G. SAARI, AND ZHIHONG XIA 229

Preface

During the last week of June, 1995, a AMS-IMS-SIAM Joint Summer Research Conference on Hamiltonian Systems and Celestial Mechanics was held at the Uni-versity of Washington in Seattle. This volume contains selective papers coming from this conference as well as a report from one of the two evening sessions devoted to "unsolved research problems." Unfortunately notes were not taken to capture the lively discussion of the other "problem session" devoted to the existence of central configurations and their properties. Partial amends, however, are made with papers in this book which offer new results about this issue.

The two main topics, Hamiltonian Systems and Celestial Mechanics, form a natural combination for a conference on dynamics because of the close symbiotic relationship they always have enjoyed. This relationship was manifested during sev-eral of the conference talks where the intellectual debt of results from area that is owed to conclusions and/or posed questions from the other became clear. Discussed topics ranged from twist maps, the Aubrey-Mather theory, Arnold diffusion, qual-itative and topological studies of systems, variational methods, to specific topics such as Melnikov's procedure and the singularity properties of particular systems. The role each area plays in stimulating the other became particularly obvious dur-ing the question section following several of the presentations and the interaction during breaks. Indeed (with apologies to Melville for badly paraphrasing his famous line), during this conference these two subjects stood hand-in-hand where a shock of recognition from one stimulated the other.

We are grateful to all the participants, particularly the speakers, for making the conference a success. Even the weather fully cooperated! We also thank Wayne Drady for his superb handling of physical arrangements during the conference and Donna Harmon and the AMS staff for their kind help during the preparation of this volume.

vii

Donald G. Saari Zhihong Xia

Hamiltonian Dynamics and Celestial Mechanics Donald G. Saari and Zhihong Xia, Editors

This book contains selected papers from the AMS-IMS-SIAM Joint Summer Research Conference on Hamiltonian Systems and Celestial Mechanics held in Seat-tle in June 1995.

The symbiotic relationship of these two topics creates a natural combination for a conference on dynamics. Topics covered include twist maps, the Aubrey-Mather theory, Arnold diffusion, qualitative and topological studies of systems, and variational methods, as well as specific topics such as Melnikov's procedure and the singularity properties of particular systems.

As one of the few books that addresses both Hamiltonian systems and celestial mechanics, this volume offers emphasis on new issues and unsolved problems. Many of the papers give new results, yet the editors purposely included some exploratory papers based on numerical computations, a section on unsolved problems, and papers that pose conjectures while developing what is known.

Features: • Open research problems • Papers on central configurations

ISBN 0-8218-0566-5

9 780821 805664