Invariant tori in the lunar problem

Abstract

Theorems on the existence of invariant KAM tori are established for perturbations of Hamiltonian systems which are circle bundle flows. By averaging the perturbation over the bundle flow one obtains a Hamiltonian system on the orbit (quotient) space by a classical theorem of Reeb. A non-degenerate critical point of the system on the orbit space gives rise to a family of periodic solutions of the perturbed system. Conditions on the critical points are given which insure KAM tori for the perturbed flow.

These general theorems are used to show that the near circular periodic solutions of the planar lunar problem are orbitally stable and are surrounded by KAM 2-tori.

For the spatial case it is shown that there are periodic solutions of two types, either near circular equatorial, that is, the infinitesimal particle moves close to the plane of the primaries following near circular trajectories and the other family where the infinitesimal particle moves along the axis perpendicular to the plane of the primaries following near rectilinear trajectories. We prove that the two solutions are elliptic and are surrounded by invariant 3-tori applying a recent theorem of Han, Li, and Yi.

In the spatial case a second averaging is performed, and the corresponding or bit space (called the twice-reduced space) is constructed. The flow of the averaged Hamiltonian on it is given and several families of invariant 3-tori are determined using Han, Li, and Yi Theorem.