LYUSTERNIK-SCHNIRELMAN THEORY FOR FLOWS AND PERIODIC-ORBITS FOR HAMILTONIAN-SYSTEMS ON T(N)XR(N)

We present a Lyusternik-Schnirelman approach for flows which generalizes the classical result concerning the number of critical points for a differentiable function on a compact manifold. This method can also be extended to special gradient-like flows on non-compact manifolds. Our main goal is the application to the existence and multiplicity of critical points for certain strongly indefinite functions of the form f: M x E --> R, where M is a compact manifold and E is a Hilbert space. The case M = T(n) of the n-dimensional torus arises in the study of periodic solutions of Hamiltonian systems which are global perturbations of completely integrable systems. For a large class of Hamiltonian systems on T*T(n) we prove the existence of at least n + 1 forced oscillations in every homotopy class of loops in T*T(n). Moreover, there exist at least n + 1 periodic solutions having an arbitrarily prescribed rational rotation vector.