, Volume 175, Issue 1, pp 1-42

The Euler-Poincaré equations and double bracket dissipation

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Abstract

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincaré) 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 to the group 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.

Research partially supported by the National Science Foundation PYI grant DMS-91-57556, and AFOSR grant F49620-93-1-0037.
Research partially supported by the AFOSR University Research Initiative Program under grants AFOSR-87-0073 and AFOSR-90-0105 and by the National Science Foundation's Engineering Research Centers Program NSFD CDR 8803012.
Research partially supported by, DOE contract DE-FG03-92ER-25129, a Fairchild Fellowship at Caltech, and the Fields Institute for Research in the Mathematical Sciences.
Research partially supported by NSF Grant DMS 91-42613, DOE contract DE-FG03-92ER-25129, the Fields Institute, the Erwin Schrödinger Institute, and the Miller Institute of the University of California.
Communicated by S.-T. Yau