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.