Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem
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| Format: | Preprint |
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2024
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| _version_ | 1866915949695729664 |
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| author | Maderna, Ezequiel Venturelli, Andrea |
| author_facet | Maderna, Ezequiel Venturelli, Andrea |
| contents | For the N-body problem we prove that any two hyperbolic rays having the same limit shape define the same Busemann function. We localize a region of differentiability for these functions, of which we know that they are viscosity solutions of the stationary Hamilton-Jacobi equation. As a first corollary, we deduce that every hyperbolic motion of the $N$-body problem must become, after some time, a calibrating curve for the Busemann function associated to its limit shape. This implies that every hyperbolic motionof the $N$-body problem is eventually a minimizer, that is, it must contain a geodesic ray of the Jacobi-Maupertuis metric. Since the viscosity solutions of the Hamilton-Jacobi equation are almost everywhere differentiable, we also deduce the generic uniqueness of geodesic rays with a given limit shape without collisions. That is to say, if the limit shape is given, then for almost every initial configuration the geodesic ray is unique. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23164 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem Maderna, Ezequiel Venturelli, Andrea Analysis of PDEs Dynamical Systems 70F10 (Primary) 70H20 37Jxx (Secondary) For the N-body problem we prove that any two hyperbolic rays having the same limit shape define the same Busemann function. We localize a region of differentiability for these functions, of which we know that they are viscosity solutions of the stationary Hamilton-Jacobi equation. As a first corollary, we deduce that every hyperbolic motion of the $N$-body problem must become, after some time, a calibrating curve for the Busemann function associated to its limit shape. This implies that every hyperbolic motionof the $N$-body problem is eventually a minimizer, that is, it must contain a geodesic ray of the Jacobi-Maupertuis metric. Since the viscosity solutions of the Hamilton-Jacobi equation are almost everywhere differentiable, we also deduce the generic uniqueness of geodesic rays with a given limit shape without collisions. That is to say, if the limit shape is given, then for almost every initial configuration the geodesic ray is unique. |
| title | Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem |
| topic | Analysis of PDEs Dynamical Systems 70F10 (Primary) 70H20 37Jxx (Secondary) |
| url | https://arxiv.org/abs/2410.23164 |