Existence of multiple periodic orbits on star-shaped hamiltonian surfaces

Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ? C²(?^2N, ?). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ? ?^2N; H(z) = c} (c > o is a constant). The surface Σ is required to verify certain geometric assumptions: Σ bounds a star-shaped compact region ? and α? ? ? ? β? for some ellipsoid ? ? ?^2N, o < α < β. We exhibit a constant δ > O (depending in an explicit fashion on the lengths of the main axes of ? and one other geometrical parameter of Σ) such that if furthermore β²/α² < 1 + δ, then (HS) has at least N distinct geometric orbits on Σ. This result is shown to extend and unify several earlier works on this subject (among them works by Weinstein, Rabinowitz, Ekeland-Lasry and Ekeland). In proving this result we construct index theories for an S¹ -action, from which we derive abstract critical point theorems for S¹ -invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.