Convexity Methods in Hamiltonian Mechanics
In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter € in the problem,
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Editorial Board
E. Bombieri, Princeton S. Feferman, Stanford N.H.Kuiper, Bures-sur-Yvette P. Lax, New York H. W. Lenstra, Jr., Berkeley R Remmert (Managing Editor), Munster W. Schmid, Cambridge, Mass. J-P. Serre, Paris 1. Tits, Paris K. K. Uhlenbeck, Austin
Ivar Ekeland
Convexity Methods in Hamiltonian Mechanics
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong
Ivar Ekeland CEREMADE, Universite Paris-Dauphine F-75775 Paris Cedex 16, France
Mathematics Subject Classification (1980): 58Exx, 58Fxx, 70Hxx, 70Jxx, 70Kxx ISBN-13: 978-3-642-74333-7 001: 10.1007/978-3-642-74331-3
e-ISBN-13: 978-3-642-74331-3
Library of Congress Cataloging-in-Publication Data Ekeland, I. (lvar), 1944- Convexity methods in Hamiltonian mechanics I Ivar Ekeland, p. cm.(Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 19) Bibliography: p. Includes index.
ISBN-13: 978-3-642-74333-7 I. Hamiltonian systems. 2. Convex domains. I. Title II. Series.
QA614.83.E44 1990 514'.74-dc20
89-11405
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© Springer-Verlag Berlin Heidelberg 1990 Softcover reprint of the hardcover 1st edition 1990 2141/3140-543210 - Printed on acid-free paper
Table of Contents
Chapter I. Linear Hamiltonian Systems ...............................
1
Floquet Theory and Stability ..................................... Krein Theory and Strong Stability ................................ Time-Dependence of the Eigenvalues of R(t) ...................... Index Theory for Positive Definite Systems ........................ The Iteration Formula ............................................ The Index of a Periodic Solution to a Nonlinear Hamiltonian System Examples ......................................................... Non-periodic Solutions: The Mean Index..........................
1 7 15 23 34 54 65 74
Chapter II. Convex Hamiltonian Systems .............................
79
1. Fundamentals of Convex Analysis .................................
2. Convex Analysis on Banach Spaces................................ 3. Integral Functionals on LOi. ..•..••...•••••.••••••••••.•...••.•...•. 4. The Clarke Duality Formula ......................................
79 86 93 98
Chapter III. Fixed-Period Problems: The Sublinear Case..............
110
Introduction ...................................................... An Existence Result .............................................. Autonomous Systems............................................. Nonautonomous Systems ......................
