The Problem of Integrable Discretization: Hamiltonian Approach
The book explores the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability
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Series Editors H. Bass J. Oesterle A. Weinstein
Yuri B. Suns
The P r o b l e m of Integrable D i s c r e t i z a t i o n : Hamiltonian Approach
Springer Basel A G
Author: Yuri B. Suris Institut für Mathematik Technische Universität Berlin Strasse des 17. Juni 136 D-10623 Berlin e-mail: [email protected]
2000 Mathematics Subject Classification 37J05, 37J15, 37J35, 37K05, 37K10, 37K15, 37K30, 35Q51, 35Q53, 35Q55, 35Q58, 37Mxx, 39A12, 70E40, 70H06
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C, USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .
ISBN 978-3-0348-9404-3 ISBN 978-3-0348-8016-9 (eBook) DOI 10.1007/978-3-0348-8016-9 This work is subject to copyright. Allrightsare reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2003 Springer Basel A G Originally published by Birkhäuser Verlag in 2003 Softcover reprint of the hardcover 1st edition 2003 Printed on acid-free paper produced of chlorine-free pulp. TCF °° ISBN 978-3-0348-9404-3 98765432 1
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Contents Preface . . . . . . . . .
XVll
I General Theory 1 Hamiltonian Mechanics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11
The problem of integrable discretization. Poisson brackets and Hamiltonian flows . Symplectic manifolds .. . . . . . . . . Poisson submanifolds and symplectic leaves Dirac bracket . . . . . Poisson reduction. . . . Complete integrability Bi-Hamiltonian systems Lagrangian mechanics on]RN Lagrangian mechanics on TP and on P x P Lagrangian mechanics on Lie groups 1.11.1 Continuous time case . . . . . . . . 1.11.2 Discrete time case . . . . . . . . . . 1.12 Invariant Lagrangians and Lie-Poisson bracket 1.12.1 Continuous time case . . . . . . . . . 1.12.2 Discrete time case . . . . . . . . . . . 1.13 Lagrangian reduction and Euler-Poincare equations . 1.13.1 Continuous time case . . . . . . . . . . . . 1.13.2 Discrete time case . . . . . . . . . . . . . . A Appendix: Gradients, vector fields, and other notation B Appendix: Lie groups and Lie algebras 1.14 Bibliographical remarks . . . . . . . .
3
4 7 11 12 14 15 18 20 21
25 28 30 35 35 37
40 40 44 47 48
49
2 R-matrix Hierarchies 2.1 2.2
Introduction . . . . . . . . . Lie-Poisson brackets . . . . . 2.2.1 General construction . 2.2.2 Tensor notation . . .
51 53 53 55
v
Contents
vi 2.2.3 Examples....... Linear r-matrix structure. . . 2.3.1 General construction. 2.3.2 Tensor notation . . . 2.3.3 Examples of R-operators and r-matrices 2.4 Generalized linear r-matrix structure 2.5 Quadratic r-matrix structure . 2.5.1 General construction. 2.5.2 Tensor notation .. . 2.5.3 Example
