Global Aspects of Classical Integrable Systems
This book gives a uniquely complete description of the geometry of the energy momentum mapping of five classical integrable systems: the 2-dimensional harmonic oscillator, the geodesic flow on the 3-sphere, the Euler top, the spherical pendulum and the La
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Global Aspects of Classical Integrable Systems Second Edition
Richard H. Cushman Larry M. Bates •
Global Aspects of Classical Integrable Systems Second Edition
Richard H. Cushman Department of Mathematics and Statistics University of Calgary Calgary Canada
Larry M. Bates Department of Mathematics and Statistics University of Calgary Calgary Canada
1st edition 1997 by Birkhäuser Verlag, Switzerland ISBN 978-3-0348-0917-7 DOI 10.1007/978-3-0348-0918-4
ISBN 978-3-0348-0918-4
(eBook)
Library of Congress Control Number: 2015939157 Mathematics Subject Classification (2010): 70-01, 70E99, 58F05, 58-01 Springer Basel Heidelberg New York Dordrecht London © Springer Basel 1997, 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com)
Table of Contents
Foreword Introduction The mathematical pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
Part I. Examples I. The harmonic oscillator 1. 2. 3. 4. 5. 6.
Hamilton’s equations and S1 symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 S1 -energy momentum mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 U(2)-momentum mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Hopf fibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Invariant theory and reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
II. Geodesics on S3 1. The geodesic and Delaunay vector fields . . . . . . . . . . . . . . . . . . . . . . . . . 2. The SO(4)-momentum mapping . . . . .
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