In this article the authors will investigate the relevance of a stable family ofrelative equilibria in a dissipative Hamiltonian system with symmetry. The authors are interested in relative equilibria of the Hamiltonian system, whose stability follows from the fact that they are local extrema of the energy-momentum function which is a combination of the Hamiltonian and a conserved quantity of the Hamiltonian system, induced by the momentum map related to the symmetry group. Although the dissipative perturbation is equivariant under the action of the symmetry group, it will destroy the conservation law associated with the symmetry group. The authors will specify its dissipative properties in terms of the induced time behavior of the momentum map and quasi-static attractive properties of the relative equilibria. By analyzing the time behavior of the previously mentioned energy-momentum function the authors derive sufficient conditions such that solutions of the dissipative system which are initially close to a relative equilibrium can be approximated by a (long) curve of relative equilibria.
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