Euler integral as a source of chaos in the three-body problem

TL;DR

This paper investigates the Euler Integral's behavior in the three-body problem, revealing chaos near the unperturbed separatrix through numerical analysis and covering relations, especially around collision singularities.

Contribution

It provides a detailed numerical study of the Euler Integral's quasi-integral property near the separatrix, highlighting chaos in the three-body problem using advanced topological methods.

Findings

Chaos detected near the unperturbed separatrix

Euler Integral shows small variations but indicates chaotic dynamics

Use of covering relations to identify chaotic regions

Abstract

In this paper we address, from a purely numerical point of view, the question, raised in [20, 21], and partly considered in [22, 9, 3], whether a certain function, referred to as "Euler Integral", is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the "unperturbed separatrix", which turns to be complicated by a collision singularity. Concretely, we reduce the Hamiltonian to two degrees of freedom and, after fixing some energy level, we discuss in detail the resulting three-dimensional phase space around an elliptic and an hyperbolic periodic orbit. After measuring the strength of variation of the Euler Integral (which are in fact small), we detect the existence of chaos closely to the unperturbed separatrix. The latter result is obtained through a careful use of the machinery of covering relations, developed in [13, 24, 23].