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 $(\mu ,\xi )$ of the relative equilibria. The dimension of the stratum of the conjugacy class (K) is $\dim G+2\dim Z(K)-\dim K$, 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.

18 pages

Cited by in corpus (4)

• Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry
• Hamiltonian Relative Equilibria with Continuous Isotropy
• $G$-isotropy of $T$-relative equilibria within manifolds tangent to spaces with linearly independent weights
• Dynamics near Relative Equilibria: Nongeneric Momenta at a 1:1 Group-Reduced Resonance