The Transversal Relative Equilibria of a Hamiltonian System with Symmetry
Abstract
We show that, given a certain transversality condition, the set of relative equilibria $\mcl E$ near $p_e\in\mcl E$ of a Hamiltonian system with symmetry 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
dimG+2dimZ(K)−dimK
, where Z(K) is the center of K, and transverse to this stratum $\mcl E$ is locally diffeomorphic to the commuting pairs of the Lie algebra of
K/Z(K)
. The stratum $\mcl E_{(K)}$ is a symplectic submanifold of P near $p_e\in\mcl E$ if and only if
p
e
is nondegenerate and K is a maximal torus of G. We also show that there is a dense subset of G-invariant Hamiltonians on P for which all the relative equilibria are transversal. Thus, generically, the types of singularities that can be found in 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.