Variational methods for periodic orbits of reduced Hamiltonian systems

Abstract

A variational method for periodic orbits is not easy to apply to a Hamiltonian system, when the symplectic form is not exact. However, if the Hamiltonian system in question is a reduced one from a Hamiltonian system on an exact symplectic manifold, the variational method applies to the latter system in order to find periodic orbits of the reduced system. This paper studies variational methods for periodic orbits in the systems reduced by the Marsden–Weinstein and the orbit reduction procedures. Periodic orbits of the reduced systems are characterized as critical points of action functionals for loops in the original phase space together with Lagrange multipliers.