Properties of Infinite Dimensional Hamiltonian Systems
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425 Paul R. Chernoff Jerrold E. Marsden
Properties of Infinite Dimensional Hamiltonian Systems
Springer-Venag Berlin · Heidelberg · New York 1974
Dr. Paul Robert Chernoff Dr. Jerrold Eldon Marsden University of California Dept. of Mathematics Berkeley, CA 94720/USA
Library of Congress Cataloging in Publication Data
Chernoff, Paul R Properties of systems.
1942dimensional Hamiltonian
in~inite
(Lecture notes in mathematics ; 425) Includes bibliographical references. 1. Hamiltonian systems. 2. Semigroups. 3. Dynamics. I. Marsden, Jerrold E., joint author. II. Title. III. Series: Lecture notes in mathematics (Berlin) ; 425. 510'.8s [516'.362] QA3.128 no. 425 [QA614.83] 74-22373
AMS Subject Classifications (1970): 34G05, 47D05, 58F05, 70D10, 70H05, 70H15 ISBN 3-540-07011-7 Springer-Verlag Berlin · Heidelberg · New York ISBN 0-387-07011-7 Springer-Verlag New York · Heidelberg · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin · Heidelberg 1974. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents
Introduction ••••••.•••••••••••••••••••.•.••••••••••••••.••.•..••••.•••••••• 1 1. Symplectic Structures and Hamiltonian Systems •.........................• 3 1.1 Strong and Weak Nondegenerate Forms •••.••••••••••••••.•.•••• 3 1. 2 Symplectic Forms •••••.•••••••••••••••••••••.•••.•.•••••••••• 4 1. 3 Canonical Symplectic Forms ...•..•...••••..•.•••............. 7 1.4 Symplectic Forms Induced by Metrics .•.•••..•.•.............. 9 1. 5 Canonical Transformations ....•......•.••.••..••..•.....•... 10 1.6 Hamiltonian Vector Fields ••.•••••.••••••••.••••.•••••.•••.• 12 1. 7 Poisson Brackets ••••••••••••••••••••••••••••••••••••••.•••• 15 2. Linear Hamil toni an Sys terns •..•........••.•...•......•...•..•........•.. 16 2.1 Introduction and Motivation--the Wave Equation •••••.••••••• l6 2.2 Canonical Commutation Relations for the Fields ••••••••••••• 22 2.3 Linear Hamiltonian Systems •.....•..••..•.....•............. 27 2. 4 Poisson Brackets and Commutators ....•.•..........•..•...... 37 2.5 Symmetry Groups and Conservation Laws .••..•.....•..•......• 38 2. 6 Complex Linear Systems ••••••••••••••••••••••..••••••••••••• 40 2.7 Complex Structure for Real Linear Systems ••••••••.•••••.••• 42 2.8 Symmetric Hyperbolic Systems ....•......•.•..•.............• 47 2.9 Some Technical Remarks: Flows of Linear Vector Fields ....• 51 3. Some General Properties of Nonlinear Semigroups, ••••••••••••••••••.•••• 56 3.1 3.2 3. 3 3.4 3.5 3. 6
Flows and Semiflows .•.•....•.....•..•..........•..•........ 56 Separate and Joint Continuity ...•••.........•..•..•........
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