2024/06/08 by Richard Montgomery, Montgomery, Richard
Engineering · Physics and Astronomy · #53 #70F #Astro and Planetary Science #Dynamical Systems (math.DS) #FOS: Mathematics #Nuclear physics research studies #Spacecraft Dynamics and Control
paper · pdf · doi:10.48550/arxiv.2406.05563
openalex publication_date 2024/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Jacobi-Maupertuis metric provides a reformulation of the classical N-body problem as a geodesic flow on an energy-dependent metric space denoted ME where E is the energy of the problem. We show that ME has finite diameter for E < 0. Consequently ME has no metric rays. Motivation comes from work of Burgos- Maderna and Polimeni-Terracini for the case E ≥ 0 and from a need to correct an error made in a previous ``proof''. We show that ME 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 ≥ 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.