Fluids and Plasmas: Geometry and Dynamics AMERICAII MATHEMATICAL SOCIETY VOLUME 28 http://dx.doi.org/10.1090/conm/028 CONTEMPORARY MATHEMATICS Titles in this Series Volume 1 Markov random fields and their applications, Ross Kindermann and J. Laurie Snell 2 Proceedings of the conference on integration, topology, and geometry in linear spaces, William H. Graves. Editor 3 The closed graph and P-closed graph properties in general topology, T. R. Hamlett and L. L. Herrington 4 Problems of elastic stability and vibrations, Vadim Komkov. Editor 5 Rational constructions of modules for simple Lie algebras, George B. Seligman 6 Umbral calculus and Hopf algebras, Robert Morris. Editor 7 Complex contour integral representation of cardinal spline functions, Walter Schempp 8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio. Editors 9 Papers in algebra, analysis and statistics, R. Lidl. Editor 10 Operator algebras and K-theory, Ronald G. Douglas and Claude Schochet. Editors 11 Plane ellipticity and related problems, Robert P. Gilbert. Editor 12 Symposium on algebraic topology in honor of Jose Adem, Samuel Gitler. Editor 1l Algebraists' homage: Papers in ring theory and related topics, S. A. Amitsur. D. J. Saltman and G. B. Seligman. Editors 14 Lectures on Nielsen fixed point theory, Boju Jiang 15 Advanced analytic number theory. Part 1: Ramification theoretic methods, Carlos J. Moreno 16 Complex representations of GL(2, K) for finite fields K, llya Piatetski-Shapiro 17 Nonlinear partial differential equations, Joel A. Smoller. Editor 18 Fix~t' points and nonexpansive mappings, Robert C. Sine. Editor 19 Proceedings of the Northwestern homotopy theory conference, Haynes R. Miller and Stewart B. Priddy. Editors 20 Low dimensional topology, Samuel J. Lomonaco. Jr .. Editor Titles in this series Volume 21 Topological methods in nonlinear functional analysis, S. P. Singh. S. Thomeier. and B. Watson. Editors 22 Factorizations of b" ± 1, b = 2, 3, 5, 6, 7,10,11,12 up to high powers, John Brillhart. D. H. Lehmer. J L. Selfridge. Bryant Tuckerman. and S. S. Wagstaff. Jr. 23 Chapter 9 of Ramanujan's second notebook-Infinite series identities, transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 24 Central extensions, Galois groups, and ideal class groups of number fields, A. Frohlich 25 Value distribution theory and its applications, Chung-Chun Yang. Editor 26 Conference in modern analysis and probability, Richard Beals. Anatole Beck. Alexandra Bellow and Arshag Hajian. Editors 27 Microlocal analysis, M. Salah Baouendi. Richard Beals and Linda Preiss Rothschild. Editors 28 Fluids and plasmas: geometry and dynamics, Jerrold E. Marsden. Editor I Volume28 Fluids and Plasmas: Geometry and Dynamics Jerrold E. Marsden1 Editor AMERICAn MATHEMATICAL SOCIETY Providence • RhOde Island EDITORIAL BOARD R. 0. Wells, Jr., Kenneth Kunen managing editor James I. Lepowsky Jeff Cheeger Johannes C. C. Nitsche Adriano M. Garsia Irving Reiner PROCEEDINGS OF THE AMS-IMS-SIAM JOINT SUMMER RESEARCH CONFERENCE IN THE MATHEMATICAL SCIENCES ON FLUIDS AND PLASMAS: GEOMETRY AND DYNAMICS HELD AT THE UNIVERSITY OF COLORADO, BOULDER JULY 17-23, 1983 These proceedings were prepared by the American Mathematical Society with partial support from the National Science Foundation Grant MCS 8218075. 1980 Mathematics Subject Classification. Primary 58Fxx, 76Exx. library of Congress Cataloging in Publication Data Main entry under title: Fluids and plasmas: Geometry and dynamics. (Contemporary mathematics, ISSN 0271·4132; v. 28) "AMS-SIAM-IMS Summer Research Conference, Boulder, Colorado, July 17-23, 1983"- lncludes bibliographies. 1. Fluid dynamics-Congresses. 2. Plasma dynamics-Congresses. 3. Differentiable dynamical systems-Congresses. 4. Geometry, Differential-Congresses. 1. Marsden, Jerrold E. II. AMS-SIAM-IMS Summer Research Conference ( 1983: Boulder, Colo.) II I. American Mathe- matical Society. IV. Society for Industrial and Applied Mathematics. V. Institute of Mathemati- cal Statistics. VI. Series: Contemporary mathematics (American Mathematical Society); v. 28. OA911.F57 1984 532'.05 84-3011 ISBN 0-8218-5028-8 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this pub- lication in reviews provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathe- matical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940. The appearance of the code on the first page of an article in this volume indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to Copyright Clearance Center, Inc. 21 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotion purposes, for creating new collective works or for resale. Copyright © 1984 by the American Mathematical Society Reprinted 1988 Printed in the United States of America All rights reserved except those granted to the United States Government This volume was printed directly from author prepared copy, The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability"§ CONTENTS Introduction . vii Conference Participants xiv Part I. Geometric-Analytic Methods A.l~einstein, Stability of Poisson-Hamilton equilibria . 3 D.Holm, J.Marsden, T.Ratiu and A.Weinstein, Stability of rigid body motion using the energy-Casimir method 15 D.Holm, Stability of planar multifluid plasma equilibria by Arno 1d's method . 25 A.Kaufman* and R.Dewar, Canonical derivation of the Vlasov-Coulomb noncanonical Poisson structure ........... 51 J.Marsden, T.Ratiu and A.Weinstein, Reduction and Hamiltonian structures on duals of semidirect product Lie algebras 55 R.Montgomery, J.Marsden and T.Ratiu, Gauged Lie-Poisson structures . 101 J.Marsden,* P.Morrison and A.Weinstein, The Hamiltonian structure of the BBGKY hierarchy equations . 115 M.Grmela, Particle and bracket formulations of kinetic equations 125 J.Marsden and P.Morrison,* Noncanonical Hamiltonian field theory and reduced MHO . 133 R.Littlejohn, Geometry and guiding center motion . 151 A.Kaufman* and B.Boghosian, Lie-transform derivation of the gyro- kinetic Hamiltonian system . 169 M.Mayer, Poisson structures for relativistic systems . 177 G.Goldin, Diffeomorphism groups, semidirect products and quantum theory. 189 Part II. Analytic and Numerical Methods N.Zabusky, Contour dynamics for two dimensional flows .. 211 Y.Wan, On the nonlinear stability of circular vortex patches . 215 T.Beale* and A.Majda, Vortex methods for fluid flow in two or three dimensions .................. 221 *An asterisk indicates the author who delivered a lecture, in case of multiple authors. v vi CONTENTS P.Olver, Hamiltonian perturbation theory and water waves ... 231 S.Wollman, Results on existence and uniqueness of solutions to the Vlasov equation ................. 251 R.Glassey and W.Strauss, Remarks on collisionless plasmas 269 H.Segur, Toward a new kinetic theory for resonant triads . 281 P.Spalart, A spectral method for external viscous flows . 315 R.Miller, Forecasting the ocean's weather: numerical models for application to oceanographic data ..... 337 Part III. Bifurcation and Dynamical Systems H.Swinney, Geometry and dynamics in experiments on chaotic systems 349 J.Guckenheimer, Dimension estimates for attractors . 357 D.Mclaughlin, J.Moloney and A.Newell,* Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity . 369 J.Crawford, Hopf bifurcation and the beam-plasma instability . 377 P.Holmes, Some remarks on chaotic particle paths in time-periodic, three-dimensional swirling flows 393 E.Siggia, A universal transition from quasi-periodicity to choas . 405 J.Curry* and E.Wayne, On the nonpathological behavior of Newton's method . 407 J.Scheurle, Successive bifurcations in the interaction of steady state and Hopf bifurcation 419 J.Swift, Convection in a rotating fluid layer ........ 435 INTRODUCTION The intention of this conference was to foster interaction among people working on mathematical, numerical and physical aspects of fluid and plasma dynamics. To this end, the organizing committee consisting of Jerry Marsden (Chairman), Philip Holmes and Andy r~.ajda, with Alex Chorin and Alan Weinstein as advisors, chose 27 speakers from the three sub-areas whom we felt would foster good interaction. We worried, though, that the conference would degenerate into three subconferences with spe- cialists talking to only themselves and leaving the audience bewildered. As it turned out, the opposite happened. The conference developed a sense of camaraderie and the speakers made every effort to bridge communi- cation gaps, despite the inevitable differences in taste and background needs that could not all be met. Another worry, prompted by our common experiences at numerous conferences, was that speakers, in their enthu- siasm, would run overtime. He offered a non-NSF