The negative energy N-body problem has finite diameter
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909219785015296 |
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| author | Montgomery, Richard |
| author_facet | Montgomery, Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The negative energy N-body problem has finite diameter Montgomery, Richard Dynamical Systems 70F, 53 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. |
| title | The negative energy N-body problem has finite diameter |
| topic | Dynamical Systems 70F, 53 |
| url | https://arxiv.org/abs/2406.05563 |