Stable Manifolds and Uniqueness of Parabolic Rays in the Newtonian $N$-Body Problem

Abstract

We study uniqueness for parabolic motions in the Newtonian $N$-body problem with a prescribed minimal normalized central configuration at infinity. A McGehee compactification identifies their tails with the strong stable manifold of the corresponding outgoing zero-energy equilibrium. The configuration projection of this manifold is locally a smooth graph, yielding a smooth initial-velocity field and uniqueness among all parabolic motions confined to a sufficiently narrow truncated cone, without any minimizing assumption. For parabolic geodesic rays, a renormalized action excess and a quantitative no-exit argument remove the confinement hypothesis when the initial configuration lies sufficiently far inside a smaller cone. We then prove that the set of collision-free initial configurations admitting more than one such ray has Lebesgue measure zero. The global argument propagates the local stable manifold backward, discards critical values of the configuration projections by Sard's theorem, and separates distinct regular branches through an explicit differential formula for their renormalized excess. No nondegeneracy assumption on the prescribed minimal central configuration is required. The proofs combine invariant-manifold theory with direct action comparisons and do not use weak KAM theory, Busemann functions, or the differentiability theory of Hamilton--Jacobi solutions.