Bifurcation of closed orbits of Hamiltonian systems with application to geodesics of the Schwarzschild metric

Abstract

We investigate bifurcation of closed orbits with a fixed energy level for a class of nearly integrable Hamiltonian systems with two degrees of freedom. More precisely, we make a joint use of Moser invariant curve theorem and Poincar\'e-Birkhoff fixed point theorem to prove that a periodic non-degenerate invariant torus $\mathcal{T}$ of the unperturbed problem gives rise to infinitely many closed orbits, bifurcating from a family of tori accumulating onto $\mathcal{T}$. The required non-degeneracy condition, which is nothing but a reformulation of the usual non-degeneracy condition in the isoenergetic KAM theory, is expressed in terms of the derivative of the apsidal angle with respect to the angular momentum: in this way, tools from the theory of time-maps of nonlinear oscillators can be used to verify it in concrete problems. Applications are given to perturbations of central force problems in the plane, and to equatorial geodesic dynamics for perturbations of the Schwarzschild metric.