Dynamics near Relative Equilibria: Nongeneric Momenta and a 1-1 Group-Reduced Resonance

Dr. George Patrick, University of Saskatchewan

The dynamics of a Hamiltonian system with symmetry near a stable relative equilibrium having non-generic momentum is to first order that of the geodesic flow of a certain metric calculated from the nilpotent part of the linearization of the relative equilibrium. This result depends on spectral splittings of the tangent space at the relative equilibrium which assume that the "group" part of the spectrum of the linearization is distinct from the "reduced" part of that spectrum. When this group-reduced nonresonance assumption is relaxed, the geodesic flow couples to a harmonic oscillator, resulting in a vastly more complicated dynamics: homoclinic connections appear, and rotational reorientation is possible, even at zero total angular momentum. I will summarize the perturbation theory behind this effect, and present some simulations illustrating it.