Qualitative features of Hamiltonian systems through averaging and reduction

Abstract

In this work we analyze the existence 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. 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. First we state results which have hypotheses on the reduced system and have conclusions about the full system. The second part is devoted to the application of the general results to the spatial lunar problem of celestial mechanics, i.e. the restricted three-body problem where the infinitesimal is close to one of the primaries. After scaling, the lunar problem is a perturbation of the Kepler problem, which after regularization is a circle bundle flow. We prove the existence of four families of periodic solutions for any small regular perturbation of the spatial Kepler problem: we find the classical near circular periodic solutions and the near rectilinear periodic solutions for all values of the small parameter. Finally we compute their approximate multipliers.