Papers by Giovanni Mancini
Communications on Pure and Applied Mathematics, 1985
Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ϵ C2(ℝ2N, ℝ). In this paper, we investi... more Consider the Hamiltonian system (HS) i = 1, …, N. Here, H ϵ C2(ℝ2N, ℝ). In this paper, we investigate the existence of periodic orbits of (HS) on a given energy surface Σ = {z ϵ ℝ2N; 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 ⊂ ℝ2N, 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 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.
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Papers by Giovanni Mancini