Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids

Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule.

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Lectures on Mechanics

Reduction in the category of Poisson manifolds is defined and some basic properties are derived.
The context is chosen to include the usual theorems on reduction of symplectic manifolds,
as well as results such as the Dirac bracket and the reduction to the Lie-Poisson bracket.

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Reduction of Poisson manifolds

Library of Congress Cataloging in Publication DataMarsden, JerroldHamiltonian Reduction by Stages(Springer Lecture Notes in Mathematics)Bibliography: p. 466Includes index.1. Hamiltonian Mechanics (Mathematics) 2. Symplectic Geometry (Math-ematics) 3. Reduction Theory (Mathematics)I. Marsden, Jerrold E. II. Misio lek, Gerard III. Title. IV. Series.QA614.A28 1983514.382-1737 ISBN 0-201-10168-SAmerican Mathematics Society (MOS) Subject Classi cation (2000): 34,37, 58, 70, 76, 93Copyright 2006 by Springer-Verlag Publishing Company, Inc.All rights reserved. No part of this publication may be reproduced, storedin a retrieval system, or transmitted, in any or by any means, electronic,mechanical, photocopying, recording, or otherwise, without the prior writ-ten permission of the publisher, Springer-Verlag Publishing Company, Inc.,New York, N.Y.

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Hamiltonian Reduction by Stages

A hamiltonian action of a Lie group G on a symplectic manifold M is generated by a momentum map J:M-*Q* which is equivariant with respect to the coadjoint representation. The reduction procedure of Meyer [14] and Marsden and Weinstein [13] consists of forming the quotient M^ =J~ (//) /G^ where /£ is an element of Q* and G^ is its coadjoint isotropy group. In recent years (see [6], for example) a property of / already known to Lie [9] has been recognized as essential: J is a Poisson map from M to g* with its Lie-Poisson structure. This suggests the problem of replacing g* by an arbitrary Poisson manifold P3 but the question immediately arises as to what object will play the role of the group G. This object having just been identified by Karasev [7] and one of us [3] [20] as a symplectic groupoid^ the purpose of the present paper is to extend the reduction procedure to symplectic groupoid actions. An important stimulus for our work has been the reduction theory for Poisson Lie group actions developed in [16]. Using Drinfel'd's notion of Poisson Lie group [4], Semenov-Tian-Shansky explained the hamiltonian behavior of the dressing transformations which arise from the inverse-scattering approach to completely integrable systems. A new theory was necessary because dressing transformations do not preserve the Poisson structure of the spaces on which they act. In a sequel to this paper5 we hope to study how Poisson Lie groups and

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Moments and Reduction for Symplectic Groupoids

Let (M, ω) be a symplectic manifold and G a Lie group acting on M by symplectic diffeomorphisms (i.e. g*ω = ω for all g ∈ G). Souriau has defined the moment of this group action as a suitable map J : M → G* (dual of the Lie algebra G of G). Some well-known results are first briefly outlined in Part I: there exists an action of G on G* for which J is equivariant, whose orbits ϑξ are symplectic manifolds (Kirillov-Souriau-Kostant’s theorem); if ξ is a regular value of J, J −1(ξ) quotiented by the action of the isotropy subgroup of ξ is, under suitable assumptions, a symplectic manifold (Meyer’s theorem).

G.-M. Marle

Papers

Symplectic Manifolds, Dynamical Groups, and Hamiltonian Mechanics