Symmetries, Topology and Resonances in Hamiltonian Mechanics
John Hornstein has written about the author's theorem on nonintegrability of geodesic flows on closed surfaces of genus greater than one: "Here is an example of how differential geometry, differential and algebraic topology, and Newton's laws make music t
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Editorial Board E. Bombieri, Princeton S. Feferman, Stanford M. Gromov, Bures-sur-Yvette H.W. Lenstra, Ir., Berkeley P.-L. Lions, Paris R. Remmert (Managing Editor), Miinster W. Schmidt, Cambridge, Mass. I.-P. Serre, Paris I. Tits, Paris
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Valerij V. Kozlov
SYlllIlletries, Topology and Resonances in Hamiltonian Mechanics
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Springer
Author Valerij V. Kozlov Department of Mathematics Moscow State University 119899 Moscow Russia Translators S.V. Bolotin Department of Mathematics Moscow State University 119899 Moscow Russia e-mail: [email protected]
D. Treshchev Department of Mathematics Moscow State University 119899 Moscow Russia e-mail: [email protected]
Yuri Fedorov B-II13 Le~in Hills Moscow State University 117234 Moscow Russia e-mail: [email protected] With 32 Figures Mathematics Subject Classification (1991): 58F05,58F07,70HXx
ISBN-13: 978-3-642-78395-1 e-ISBN-13: 978-3-642-78393-7 DOl: 10.1007/978-3-642-78393-7 Library of Congress Cataloging-in-Publication Data. Kozlov, V.V. (Valerif Viktorovich) Symmetries, topology, and resonances in Hamiltonian mechanics I Valery V. Kozlov. p. cm. Includes index. ISBN 3-540-57039-X - ISBN 0-387-57039-X I. Hamiltonian systems. 2. Symmetry (physics) 3. Topology. 4. Mathematical physics. I. Title. QC20.7.H35K69 1995 531'.1 1'01 5 I 474-ilc20 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, boardcasting, reproduction on microfilms or in any other ways, and store in data banks. Duplication of this publication or parts of thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996
Softcover reprint of the hardcover 1st edition 1996
Typeset with lEX: Data conversion by Lewis & Leins GmbH, Berlin SPIN: 10015730 4113020 - 5 4 3 2 I 0 - Printed on acid-free paper
Mephistopheles Ich wiinschte nicht, Euch irrezufiihren. Was die Wissenschaft betrifft, Es ist so schwer, den falschen Weg zu meiden, Es liegt in ihr so viel verborgenes Gift, Und von der Arzenei ists kaum zu unterscheiden. Am besten ists auch hier, wenn Ihr nur Einen hart Und auf des Meisters Worte schwart. 1m ganzen: haltet Euch an Worte ! Dann geht Ihr durch die sichre Pforte Zum Tempel der GewiGheit ein. Schiller Doch ein Begriff muG bei dem Worte sein.
Faust I. Goethe. Goethe Werke B. 3. Insel Verlag. Frankfurt am Main 1977
Preface
The problem of exact integration of equations of dynamics has been one of the most popular fields of research since Newton's famous "Philosophiae naturalis principia mathematic a" . The principal idea in the investigation is the general notion of symmetry. When solving the problem of a mass point motion in a central field, Newton already used the concept
