CaltechAUTHORS
  A Caltech Library Service

The Euler-Poincare Equations and Double Bracket Dissipation

Bloch, Anthony and Krishnaprasad, P. S. and Marsden, Jerrold E. and Ratiu, Tudor S. (1996) The Euler-Poincare Equations and Double Bracket Dissipation. Communications in Mathematical Physics, 175 (1). pp. 1-42. ISSN 0010-3616. http://resolver.caltech.edu/CaltechAUTHORS:20100713-150040832

[img]
Preview
PDF - Updated Version
See Usage Policy.

332Kb

Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20100713-150040832

Abstract

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincare) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad,Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/BF02101622DOIUNSPECIFIED
http://www.springerlink.com/content/w045l956w3176486/?p=3c69af060a424c04be50423233d4b9b8&pi=0PublisherUNSPECIFIED
Additional Information:© 1996. March, 1993; this version, June 4, 1996. Received: 11 January 1994 Revised: 23 November 1994. Communicated by S.-T. Yau. We thank Miroslav Grmela, Darryl Holm, Alan Kaufman, Naomi Leonard, Peter Michor, Gloria Sanchez and the referees for helpful suggestions. We also thank the Fields Institute for providing the opportunity to meet in pleasant surroundings during which time some of the ideas in the paper were first worked out. We also thank the Erwin Schrödinger Institute for Mathematical Physics for their hospitality.
Funders:
Funding AgencyGrant Number
NSF/PYI DMS-91-57556
Air Force Office of Scientific Research (AFOSR)F49620-93-1-0037
University Research Initiative ProgramAFOSR-87-0073
University Research Initiative ProgramAFOSR-90-0105
NSF Engineering Research Centers ProgramCDR 8803012
Department of EnergyDE-FG03-92ER-25129
Fairchild Fellowship at CaltechUNSPECIFIED
Fields Institute for Research in the Mathematical SciencesUNSPECIFIED
NSFDMS 91-42613
Department of Energy DE-FG03-92ER-25129
Fields InstituteUNSPECIFIED
Erwin Schrödinger InstituteUNSPECIFIED
Miller Institute of the University of CaliforniaUNSPECIFIED
Record Number:CaltechAUTHORS:20100713-150040832
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100713-150040832
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19039
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:04 Aug 2010 17:38
Last Modified:26 Dec 2012 12:14

Repository Staff Only: item control page