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<p>For the <i>N</i>-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 <i>N</i>-body problem must become, after some time, a calibrating curve for the Busemann function associated to its limit shape. This implies that every hyperbolic motion of the <i>N</i>-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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年1月12日

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  • Ezequiel Maderna,
  • Andrea Venturelli

摘要

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 motion of 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.