Global hypersurfaces of section in the spatial restricted three-body problem

TL;DR

This paper introduces a contact-topological method to establish the existence of global hypersurfaces of section in the spatial restricted three-body problem, providing a non-perturbative framework applicable across different mass ratios.

Contribution

It constructs global hypersurfaces of section for the spatial restricted three-body problem using contact topology, extending previous results to a broader energy range and including the integrable Kepler case.

Findings

Existence of a circle family of global hypersurfaces of section below the first critical energy.

Hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the 2-sphere.

First return map is Hamiltonian and relates to the Reeb flow in the planar problem.

Abstract

We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of section for the regularized dynamics. Below the first critical value, these hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the $2$-sphere, and they carry symplectic forms on their interior, which are each deformation equivalent to the standard symplectic form. The boundary of the global hypersurface of section is an invariant set for the regularized dynamics that is equal to a level set of the Hamiltonian describing the regularized planar problem. The first return map is Hamiltonian, and restricts to the boundary as the time-$1$ map of a positive reparametrization of the Reeb flow in the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.