THE EULER-POINCARE EQUATIONS AND DOUBLE BRACKET DISSIPATION

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincare) system by a special dissipation term that has Broc kett's double bracket form. We show that a formally unstable equilibri um of the unperturbed system becomes a spectrally and hence nonlinearl y unstable equilibrium after the perturbation is added. We also invest igate the geometry of this dissipation mechanism and its relation to R ayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we stu died the corresponding problem for systems with symmetry with the diss ipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here incl ude a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for diss ipative rigid body dynamics and geophysical fluids, and certain relati ve equilibria in plasma physics and stellar dynamics.