Chazy-Type Asymptotics and Hyperbolic Scattering for the $n$-Body Problem

TL;DR

This paper analyzes hyperbolic solutions in the Newtonian n-body problem, providing new proofs of classical asymptotics, studying the structure of solutions at infinity, and exploring scattering relations between asymptotic states.

Contribution

It introduces an analytical linearization of the flow near infinity and offers new insights into the scattering problem for hyperbolic solutions in the n-body problem.

Findings

Flow near hyperbolic infinity can be analytically linearized.

Provides a new proof of Chazy's asymptotic formulas.

Studies scattering relations for solutions hyperbolic in both time directions.

Abstract

We study solutions of the Newtonian $n$-body problem which tend to infinity hyperbolically, that is, all mutual distances tend to infinity with nonzero speed as $t \rightarrow +\infty$ or as $t \rightarrow -\infty$. In suitable coordinates, such solutions form the stable or unstable manifolds of normally hyperbolic equilibrium points in a boundary manifold "at infinity". We show that the flow near these manifolds can be analytically linearized and use this to give a new proof of Chazy's classical asymptotic formulas. We also address the scattering problem, namely, for solutions which are hyperbolic in both forward and backward time, how are the limiting equilibrium points related? After proving some basic theorems about this scattering relation, we use perturbations of our manifold at infinity to study scattering "near infinity", that is, when the bodies stay far apart and interact only weakly.