Integrable Systems in Celestial Mechanics
This work presents a unified treatment of three important integrable problems relevant to both Celestial and Quantum Mechanics. Under discussion are the Kepler (two-body) problem and the Euler (two-fixed center) problem, the latter being the more complex
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Editors-in-Chief Anne Boutet de Monvel, Universit´ e Paris VII Denis Diderot Gerald Kaiser, Center for Signals and Waves, Austin, TX
Editorial Board C. Berenstein, University of Maryland, College Park Sir M. Berry, University of Bristol P. Blanchard, Universit¨ at Bielefeld M. Eastwood, University of Adelaide A.S. Fokas, University of Cambridge C. Tracy, University of California, Davis
´ Math´ Diarmuid O una
Integrable Systems in Celestial Mechanics
Birkh¨ auser Boston • Basel • Berlin
´ Math´ Diarmuid O una Dublin Institute of Advanced Studies 10, Burlington Road Dublin, 4. Ireland [email protected]
ISBN-13: 978-0-8176-4096-5 DOI: 10.1007/978-0-8176-4595-3
e-ISBN-13: 978-0-8176-4595-3
Library of Congress Control Number: 2008925448 Mathematics Subject Classification (2000): 34A05, 70-XX, 70-02, 70F05, 70F15, 70H06, 70M20, 85-02 c 2008 Birkhauser ¨ Boston, a part of Springer Science + Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨ auser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
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I gcuimhne ar mo thuismitheoir´ı (To the memory of my parents)
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 0 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Lagrangian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ignorable Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Separable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Liouville Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21 21 22 24 26
2 The Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Features of the Ellipse: Geometry and Analysis . . . . . . . . . . . . . . . . . . . . . 2 The Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Kepler Problem: Vectorial Treatmen
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