Stability of geodesic-ray data, horofunctions, and rectifiability of fixed-shape slices in the Newtonian \(N\)-body problem

TL;DR

This paper investigates the stability and geometric structure of solution sets in the Newtonian N-body problem, revealing compactness properties, rectifiability, and dimension of fixed-shape slices in phase space.

Contribution

It establishes a stability theorem for geodesic rays, describes the rectifiability and Hausdorff dimension of fixed-shape slices, and links phase-space compactness with geometric measure theory.

Findings

Limits of geodesic rays remain collision-free and generate geodesic rays.

Fixed-shape slices are countably rectifiable with Hausdorff dimension exactly d(N-1).

Normalized Busemann functions converge to viscosity solutions of the stationary equation.

Abstract

For the Newtonian \(N\)-body problem at nonnegative energy, we study solution sets selected by the Jacobi--Maupertuis variational principle and by the associated stationary Hamilton--Jacobi equation. We prove a compactness/stability theorem for classical initial data generating geodesic rays: limits in the ambient phase space remain collision-free, generate geodesic rays, and carry locally relatively compact normalized Busemann functions. The limiting horofunction of normalized Busemann functions yields a viscosity solution of the limiting stationary equation. For a fixed collision-free hyperbolic limit shape \(a\), we also prove closedness of the corresponding slice of geodesic-ray data. Finally, after passing to the reduced configuration space \(X\), we show that this fixed-shape slice is countably \(d(N-1)\)-rectifiable in phase space and has Hausdorff dimension exactly \(d(N-1)\). Thus the paper combines phase-space compactness of calibrated minimizing motions with a geometric-measure description of a fixed-shape Hamilton--Jacobi calibrated slice.