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

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.

💡 Research Summary

The paper investigates the long‑time behavior of solutions to the Newtonian N‑body problem with positive energy, focusing on the relationship between hyperbolic motions, Busemann functions, and the Jacobi–Maupertuis metric. A hyperbolic motion is defined as a trajectory x(t) that satisfies x(t)=t a+o(t) as t→∞, where a∈Ω is a collision‑free configuration and the associated energy is h=½‖a‖²>0. The authors first recall the variational formulation: the Lagrangian L(x, v)=½‖v‖²+U(x) and the Hamiltonian H(x, p)=½‖p‖²−U(x). For any fixed energy h≥0 the Jacobi–Maupertuis distance ϕ_h between two configurations is the minimal action of L+h without a time constraint, and it induces a Riemannian metric j_h on the collision‑free configuration space Ω. With this metric (Ω, j_h) becomes a geodesic space, and hyperbolic motions are precisely geodesic rays for j_h when they are parametrized so that H(x, ẋ)=h.

The central object of study is the Busemann function associated with a geodesic ray γ, defined as the limit b_γ(x)=lim_{t→∞}