Relative equilibria and stability in reduced presymplectic Hamiltonian systems

Study the relative equilibria of the Hamiltonian vector field associated with the homogeneous function induced by a Hamiltonian section, together with their stability, in order to extend the reduction theory to the excluded relative-equilibrium points.

Background

The reduction procedure developed in the paper requires excluding points at which the Hamiltonian vector field of the homogeneous extended Hamiltonian lies in the tangent space to a symmetry orbit. These points are precisely the relative equilibria of the Hamiltonian vector field. Their exclusion ensures that the reduced presymplectic forms have the expected coranks two and one on the extended and restricted reduced phase spaces, respectively.

The paper notes that the image under the principal-bundle projection of a relative equilibrium for the extended Hamiltonian vector field is a relative equilibrium for the time-dependent Hamiltonian dynamics. A systematic analysis of these relative equilibria and their stability would therefore clarify the behavior of the reduction at the points omitted from the present theory.