The negative energy N-body problem has finite diameter

TL;DR

This paper proves that the Jacobi-Maupertuis metric space for the negative energy N-body problem has finite diameter, implying no infinite geodesic rays exist, and corrects previous misconceptions about the case when energy is non-negative.

Contribution

It establishes the finite diameter of the Jacobi-Maupertuis metric space for negative energy, correcting earlier errors and extending understanding of the N-body problem's geometric properties.

Findings

M_E has finite diameter for E < 0

No metric rays in M_E for E < 0

Provides a method to quantify escape rates in configuration space

Abstract

The Jacobi-Maupertuis metric provides a reformulation of the classical N-body problem as a geodesic flow on an energy-dependent metric space denoted $M_E$ where $E$ is the energy of the problem. We show that $M_E$ has finite diameter for $E < 0$. Consequently $M_E$ has no metric rays. Motivation comes from work of Burgos- Maderna and Polimeni-Terracini for the case $E \ge 0$ and from a need to correct an error made in a previous ``proof''. We show that $M_E$ has finite diameter for $E < 0$ by showing that there is a constant $D$ such that all points of the Hill region lie a distance $D$ from the Hill boundary. (When $E \ge 0$ the Hill boundary is empty.) The proof relies on a game of escape which allows us to quantify the escape rate from a closed subset of configuration space, and the reduction of this game to one of escaping the boundary of a polyhedral convex cone into its interior.