Periodic Solutions of the N-Body Problem
The N-body problem is the classical prototype of a Hamiltonian system with a large symmetry group and many first integrals. These lecture notes are an introduction to the theory of periodic solutions of such Hamiltonian systems. From a generic point of vi
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1719
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1719
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Kenneth R. Meyer
Periodic Solutions of the N-Body Problem
Springer
Author Kenneth R. Meyer Department of Mathematics University of Cincinnati Cincinnati, Ohio 45221-0025 USA E-mail: [email protected]
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Meyer, Kenneth R.: Periodic solutions of the N-bodv problem / Kenneth R. Meyer. Berlin; Heidelberg; New York' ; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 (Lecture notes in mathematics; 1719) ISBN 3-540-66630-3
Mathematics Subject Classification (1991): 58F05, 58F22, 70FIO, 70F 15 ISSN 0075-8434 ISBN 3-540-66630-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer- Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10700296 41/3143-543210 - Printed on acid-free paper
Preface
These notes grew out of a series of lectures that I gave at the Universidade Federal de Pernambuco, Recife, Brazil. Since this was a limited number of lectures in the extensive area of periodic solutions of the N-body problem, I was forced to define a small subset of the literature in order to give a reasonably complete introduction. Filling in the most of the details resulted in these lecture notes. From a generic point of view the N-body problem is highly degenerate. It is invariant under the symmetry group of Euclidean motions and admits linear momentum, angular momentum and energy as integrals. This implies that an attempt to apply the implicit function directly yields a Jacobian with nullity 8 for the planar problem and nullity 12 for the spatial problem. (The multiplier + 1 has multiplicity 8 in the planar problem and 12 in the spatial problem.) Therefore, the integrals and symmetries must be confronted head on, which leads to the definition of the reduced space where all the known integrals and symmetries have been eliminated. It is on the reduced space that one can hope for a nonsingular Jacobian without
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