Existence of multiple periodic orbits on star‐shaped hamiltonian surfaces

Abstract

Abstract Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ϵ C 2 (ℝ 2N , ℝ). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = { z ϵ ℝ 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 ℛ and αℰ ⊂ ℛ ⊂ βℰ for some ellipsoid ℰ ⊂ ℝ 2N , o < α < β. We exhibit a constant δ > O (depending in an explicit fashion on the lengths of the main axes of ℰ and one other geometrical parameter of Σ) such that if furthermore β 2 /α 2 < 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 S 1 ‐action, from which we derive abstract critical point theorems for S 1 ‐invariant functionals. We also derive an estimate for the minimal period of solutions to differential equatious.