The Euler-Poincare Equations and Double Bracket Dissipation
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.
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.Attached Files
Updated - BlKrMaRa1996.pdf
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Additional details
- Eprint ID
- 19039
- Resolver ID
- CaltechAUTHORS:20100713-150040832
- DMS-91-57556
- NSF
- F49620-93-1-0037
- Air Force Office of Scientific Research (AFOSR)
- AFOSR-87-0073
- Air Force Office of Scientific Research (AFOSR)
- AFOSR-90-0105
- Air Force Office of Scientific Research (AFOSR)
- CDR 8803012
- NSF
- DE-FG03-92ER-25129
- Department of Energy (DOE)
- Sherman Fairchild Foundation
- Fields Institute for Research in the Mathematical Sciences
- DMS 91-42613
- NSF
- DE-FG03-92ER-25129
- Department of Energy (DOE)
- Erwin Schrödinger Institute
- Miller Institute for Basic Research in Science
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2010-08-04Created from EPrint's datestamp field
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2021-11-08Created from EPrint's last_modified field