Introduction to the Perturbation Theory of Hamiltonian Systems
This book presents the basic methods of regular perturbation theory of Hamiltonian systems, including KAM-theory, splitting of asymptotic manifolds, the separatrix map, averaging, anti-integrable limit, etc. in a readable way. Although concise, it discuss
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Dmitry Treschev Oleg Zubelevich
1 Introduction to the Perturbation Theory
Introduction to the Perturbation Theory of Hamiltonian Systems
13
Springer Monographs in Mathematics
For other titles published in this series, go to http://www.springer.com/series/3733
Dmitry Treschev Oleg Zubelevich
Introduction to the Perturbation Theory of Hamiltonian Systems
Dmitry Treschev Department of Mechanics Russian Academy of Sciences Steklov Mathematical Institute ul. Gubkina 8 Moscow 119991 Russia [email protected]
Oleg Zubelevich Department of Mechanics and Mathematics Moscow State University Leninskie Gory Moscow 119899 Russia [email protected]
ISSN 1439-7382 ISBN 978-3-642-03027-7 e-ISBN 978-3-642-03028-4 DOI 10.1007/978-3-642-03028-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009937641 Mathematics Subject Classification (2000): 37JXX, 37J05, 37J10, 37J25, 37J30, 37J40, 37J45 ©Springer-Verlag Berlin Heidelberg 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. 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. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
This book is an extended version of lectures given by the first author in 1995–1996 at the Department of Mechanics and Mathematics of Moscow State University. We believe that a major part of the book can be regarded as an additional material to the standard course of Hamiltonian mechanics. In comparison with the original Russian version1 we have included new material, simplified some proofs and corrected misprints. Hamiltonian equations first appeared in connection with problems of geometric optics and celestial mechanics. Later it became clear that these equations describe a large class of systems in classical mechanics, physics, chemistry, and other domains. Hamiltonian systems and their discrete analogs play a basic role in such problems as rigid body dynamics, geodesics on Riemann surfaces, quasi-classic approximation in quantum mechanics, cosmological models, dynamics of particles in an accelerator, billiards and other systems with elastic reflections, many infinite-dimensional models in mathematical physics, etc. In this book we study Hamiltonian systems assuming that they depend on some par
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