# The Euler-Poincaré Equations and Double Bracket Dissipation

**Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden
**

*Comm. Math. Phys.*, **175**, 1-42

### 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.