The transversal relative equilibria of a Hamiltonian system with symmetry

Let P be a symplectic manifold with a free symplectic action of a connected compact Lie group G. We show that, given a certain transversality condition, the set of relative equilibria E nearpe ∈ E of aG-invariant Hamiltonian system onP is locally Whitney-stratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentum-generator pairs (μ, ξ) of the relative equilibria. The dimension of the stratum of the conjugacy class (K) is dim G+2 dim Z(K)−dim K , whereZ(K) is the centre ofK . Transverse to this stratum E is locally diffeomorphic to the set of commuting pairs of the Lie algebra of K/Z(K). The stratum E(K) is a symplectic submanifold of P near pe ∈ E if and only if pe is non-degenerate and K is a maximal torus of G. We also show that the set of G-invariant Hamiltonians on P for which all the relative equilibria are transversal is open and dense. Thus, generically, the types of singularities of the set of relative equilibria of a Hamiltonian system with symmetry are those types found amongst the singularities at zero of the sets of commuting pairs of certain Lie subalgebras of the symmetry group. AMS classification scheme numbers: 58F05, 70H33, 58F14