Computational symplectic topology and symmetric orbits in the restricted three-body problem

TL;DR

This paper introduces a computational approach combining validated numerics and symplectic topology to prove the existence of certain periodic orbits and surfaces of section in the restricted three-body problem, advancing the proof of Birkhoff's conjecture.

Contribution

It develops methods to rigorously compute the Conley-Zehnder index and demonstrates the existence of approximate circular orbits, providing initial steps toward a computational proof of Birkhoff's conjecture.

Findings

Existence of approximately circular direct orbits for various parameters

Methods for rigorous computation of Conley-Zehnder index

Application to symmetric orbits in the three-body problem

Abstract

In this paper we propose a computational approach to proving the Birkhoff conjecture on the restricted three-body problem, which asserts the existence of a disk-like global surface of section. Birkhoff had conjectured this surface of section as a tool to prove existence of a direct periodic orbit. Using techniques from validated numerics we prove the existence of an approximately circular direct orbit for a wide range of mass parameters and Jacobi energies. We also provide methods to rigorously compute the Conley-Zehnder index of periodic Hamiltonian orbits using computational tools, thus giving some initial steps for developing computational Floer homology and providing the means to prove the Birkhoff conjecture via symplectic topology. We apply this method to various symmetric orbits in the restricted three-body problem.