Families of Relative Equilibria in Hamiltonian Systems with Dissipation

In this note the influence of dissipation on families of relative equilibria in Hamiltonian systems will be considered. Relative equilibria can be described as critical points of an appropriate functional. This characterisation can be used to give sufficient conditions such that in finite dimensional systems with dissipation the extremal families of relative equilibria are stable under dissipation. Furthermore, a full class of families of relative equilibria in the Navier-Stokes equations will be analysed. For these families it will be shown that the extremal family of relative equilibria is an attractor and the non-extremal families of relative equilibria are unstable.