Global Surfaces of Section and Periodic Orbits in The Spatial Isosceles Three Body Problem

TL;DR

This paper investigates the spatial isosceles three body problem, establishing the existence of global surfaces of section and proving the abundance of periodic orbits for generic mass choices using advanced index theory.

Contribution

It introduces a method to find global surfaces of section with specific boundary orbits and proves the existence of infinitely many periodic orbits in the spatial isosceles three body problem.

Findings

Existence of disk-like global surfaces of section with Euler orbit boundary

Proof of infinitely many periodic orbits for generic masses

Formulas relating rotation numbers to mean indices via Maslov index

Abstract

We study the spatial isosceles three body problem, which is a system with two degrees of freedom after modulo the rotation symmetry. For certain choices of energy and angular momentum, we find some disk-like global surfaces of section with the Euler orbit as their common boundary, and a brake orbit passing through them. By considering the Poincar\'e maps of these global surfaces of section, we prove the existence of all kinds of different periodic orbits under certain assumption. Moreover, we are able to prove, for generic choices of masses, the system always has infinitely many periodic orbits. One of the key is to estimate the rotation numbers of the Euler orbit and the brake orbit with respect to the Poincar\'e map. For this, we establish formulas connected these numbers with the mean indices of the corresponding orbits using the Maslov-type index.