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 [7] and by Lerman and Singer [3]. 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. Introduction Many Hamiltonian systems arising in nature possess symmetry and in particular continuous symmetry—most commonly a group of rotations or Euclidean motions, whether in the plane or in space. In this note we consider relative equilibria in such systems, which are motions that coincide with 1-parameter symmetry transformations. Given such a relative equilibrium, it is often important to decide on its (nonlinear) stability, and there are criteria for determining this based on Dirichlet’s criterion for ordinary equilibria, but involving the velocity of the relative equilibrium through an appropriate element of the Lie algebra g of the group G. If the action is locally free at the relative equilibrium (meaning the isotropy subgroup at any point of the relative equilibrium is finite) then the “relative Dirichlet criterion” is straightforward because the velocity corresponds to a unique element of the Lie algebra g. However, when the action fails to be locally free the story is less clear because there will be many different “group velocities” for a given physical velocity. In the late 1990s two papers were published, by Ortega and Ratiu [7] and Lerman and Singer [3], adapting the Dirichlet criterion to deal with this case, while a paper by the first author [4] provides a more topological criterion, that of an “extremal relative equilibrium”, which we will use in the proof below. The method of Ortega-Ratiu and Lerman-Singer involves having a splitting of the Lie algebra g, and showing there is a unique preferred group velocity relative to this splitting, which they call the ‘orthogonal velocity’, and then using this orthogonal velocity to define a relative Dirichlet criterion, analogous to the locally free case. Moreover, Ortega and Ratiu give an example showing how different choices of splitting may produce different critieria for stability so it may be necessary to consider all possible splittings. The purpose of this note is to dispense with the splitting construction, and to show that the relative Dirichlet criterion is sufficient to guarantee stability, using any group veclocity, not merely those that arise from a splitting. In the special case that the relative equilibrium is an equilibrium, the orthogonal