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arXiv · 2410.23164

Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem

Ezequiel Maderna·Andrea Venturelli
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Abstract

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.

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Ezequiel Maderna, Andrea Venturelli. 2025-11-13. Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem. https://doi.org/10.1007/s00526-025-03225-w

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