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

TL;DR

This paper studies the bifurcation of closed orbits in nearly integrable Hamiltonian systems, applying advanced theorems to demonstrate the emergence of infinitely many orbits near a known invariant torus, with applications to geodesics in Schwarzschild spacetime.

Contribution

It introduces a novel non-degeneracy condition based on the derivative of the apsidal angle, linking KAM theory with nonlinear oscillator time-maps, and applies it to geodesic dynamics in Schwarzschild metrics.

Findings

Infinitely many closed orbits bifurcate from invariant tori.

The non-degeneracy condition can be verified via nonlinear oscillator theory.

Applications to perturbed central force problems and Schwarzschild geodesics.

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.