Tadashi Tokieda - Perturbation theory for symmetric hamiltonian systems

The problem of persistence, bifurcation, stability of periodic orbits and equilibria is of special importance in perturbation theory, and many results are known (e.g. Poincaré, Weinstein, Moser). These results generalize to systems that have symmetries given by a hamiltonian action of a Lie group. The natural objects of study are relative periodic orbits (orbits that are closed up to group action) and relative equilibria. We derive a recipe for reducing the theory to the classical theory, which works even in the hard case when we reduce at a singular value of the moment map (as we often must in real life). Plane vortices are discussed as examples; moreover, some of their exact solutions show that the hypothesis in our recipe is tight. [joint work with Lerman and Montaldi]