Mathematics – Symplectic Geometry
Scientific paper
1999-02-16
Mathematics
Symplectic Geometry
18 pages
Scientific paper
10.1088/0951-7715/13/6/311
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.
Patrick George W.
Roberts Randy Mark
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