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

TL;DR

This paper proves the uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem and shows that hyperbolic motions eventually become minimizers associated with their limit shape.

Contribution

It establishes the uniqueness of Busemann functions for hyperbolic rays and links hyperbolic motions to minimizers in the N-body problem, with generic uniqueness results.

Findings

Hyperbolic rays with the same limit shape define the same Busemann function.

Hyperbolic motions eventually become calibrating curves for their Busemann functions.

Almost every initial configuration yields a unique geodesic ray for a given limit shape.

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.