Persistence and Stability of Relative Equilibria

We consider relative equilibria in symmetric Hamiltonian systems, and their persistence or bifurcation as the momentum is varied. In particular, we extend a classical result about persistence of relative equilibria from values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either (i) the relative equilibrium is at a local extremum of the reduced Hamiltonian, or (ii) the action on the phase space is (locally) free. The first case uses just point-set topology, while in the second we rely on the local normal form for (free) symplectic group actions, and then apply the splitting lemma. We also consider the Lyapunov stability of extremal relative equilibria. The group of symmetries is assumed to be compact.