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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.

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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.

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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.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)F49620-93-1-0037
Air Force Office of Scientific Research (AFOSR)AFOSR-87-0073
Air Force Office of Scientific Research (AFOSR) AFOSR-90-0105
NSFCDR 8803012
Department of Energy (DOE)DE-FG03-92ER-25129
Sherman Fairchild FoundationUNSPECIFIED
Fields Institute for Research in the Mathematical SciencesUNSPECIFIED
NSFDMS 91-42613
Department of Energy (DOE)DE-FG03-92ER-25129
Erwin Schrödinger InstituteUNSPECIFIED
Miller Institute for Basic Research in ScienceUNSPECIFIED
Record Number:CaltechAUTHORS:20100713-150040832
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19039
Deposited By: Ruth Sustaita
Deposited On:04 Aug 2010 17:38
Last Modified:01 May 2015 19:51

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