APPROXIMATIONS WITH CURVES OF RELATIVE EQUILIBRIA IN HAMILTONIAN-SYSTEMS WITH DISSIPATION

In this paper we will investigate the relevance of a stable family of relative equilibria in a dissipative Hamiltonian system with symmetry. We are interested in relative equilibria of the Hamiltonian system, w hose stability follows from the fact that they are local extrema of th e energy-momentum function which is a combination of the Hamiltonian a nd a conserved quantity of the Hamiltonian system, induced by the mome ntum map related to the symmetry group. Although the dissipative pertu rbation is equivariant under the action of the symmetry group, it will destroy the conservation law associated with the symmetry group. We w ill specify its dissipative properties in terms of the induced time be haviour of the momentum map and quasi-static attractive properties of the relative equilibria. By analysing the time behaviour of the previo usly mentioned energy-momentum function we derive sufficient condition s such that solutions of the dissipative system which are initially cl ose to a relative equilibrium can be approximated by a (long) curve of relative equilibria. At the end we illustrate the method by analysing the example of a rigid body in a rotational symmetric field with diss ipative rotation-like perturbation added.