Classical $n$-body scattering with long-range potentials

TL;DR

This paper studies classical n-body scattering with long-range pair potentials, establishing a global surface of section and smooth scattering map, advancing understanding of long-range interactions in classical mechanics.

Contribution

It introduces an explicit global surface of section for long-range n-body scattering and proves the smoothness of the scattering map, linking dynamics to free motion.

Findings

Existence of a global surface of section for the free region.

Smoothness of the scattering map.

Forward conjugacy between n-body and free dynamics.

Abstract

We consider the scattering of $n$ classical particles interacting via pair potentials, under the assumption that each pair potential is "long-range", i.e. being of order ${\cal O}(r^{-\alpha})$ for some $\alpha >0$. We define and focus on the "free region", the set of states leading to well-defined and well-separated final states at infinity. As a first step, we prove the existence of an explicit, global surface of section for the free region. This surface of section is key to proving the smoothness of the map sending a point to its final state and to establishing a forward conjugacy between the $n$-body dynamics and free dynamics.