Periodic Solutions in Hamiltonian Systems, Averaging, and the Lunar Problem

Abstract

We investigate the existence, characteristic multipliers, and stability of periodic solutions to a Hamiltonian vector field which is a small perturbation of a vector field tangent to the fibers of a circle bundle. Our primary examples are the planar lunar and spatial lunar problems of celestial mechanics, i.e., the restricted three-body problem where the infinitesimal is close to one of the primaries. By averaging the perturbation over the fibers of the circle bundle one obtains a Hamiltonian system on the reduced (orbit) space of the circle bundle. Our goal in the first part of the paper is to state and prove results which have hypotheses on the reduced system and have conclusions about the full system. Starting with the classical work of Reeb, we give a summary of lemmas, corollaries, and theorems about the existence, characteristic multipliers, and stability of periodic solutions to Hamiltonian systems which are perturbations of circle bundle flows. By reformulating the classical results in modern language and giving alternative proofs in place of the original proofs, we are able to infer new consequences of these classical results. The second part of the paper is devoted to applications of the general results. We apply these general results to the planar and spatial lunar problem. After scaling, the lunar problem is a perturbation of the Kepler problem, which after regularization is a circle bundle flow. We find the classical near-circular periodic solutions and the near-rectilinear periodic solutions. Then we compute their approximate multipliers and show that there is a “twist.” However, the twist is too degenerate to apply the classical KAM theorem on invariant tori. We also find symmetric periodic solutions which are continuations of elliptic solutions of the Kepler problem.