On the stability of Hamiltonian relative equilibria with non-trivial isotropy

We consider Hamiltonian systems with symmetry, and relative equilibria with isotropy subgroup of positive dimension. The stability of such relative equilibria has been studied by Ortega and Ratiu (1999 Nonlinearity 12 693–720) and by Lerman and Singer (1998 Nonlinearity 11 1637–49). In both papers the authors give sufficient conditions for stability which require first determining a splitting of a subalgebra of g, with different splittings giving different criteria. In this note we remove this splitting construction and so provide a more general and more straightforward criterion for stability. The result is also extended to apply to systems whose momentum maps are not coadjoint equivariant