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

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 $t\to +\infty$ or as $t\to -\infty$. 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.

Minor changes compare with previous version. To appear in ARMA

References in corpus (2)

• Free time minimizers for the planar three-body problem
• An Index Theory for Collision, Parabolic and Hyperbolic Solutions of the Newtonian $n$-body Problem

Cited by in corpus (6)

• Geodesic rays of the $N$-body problem
• On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields
• Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions
• An Index Theory for Collision, Parabolic and Hyperbolic Solutions of the Newtonian $n$-body Problem
• Uniqueness of hyperbolic Busemann functions in the Newtonian N-body problem
• Classical $n$-body scattering with long-range potentials