The Euler - Poincar e Equations

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincar e) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the un-perturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of 1 this dissipation mechanism and its relation to Rayleigh dissipation functions. 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 uids, and certain relative equilibria in plasma physics and stellar dynamics.