Improbability of Collisions in $n$-Body Systems

TL;DR

This paper proves that in many two-body interaction systems like gravity or electrostatics, collision trajectories are extremely rare, occupying zero measure in phase space, using advanced symplectic geometry methods.

Contribution

It introduces a novel geometric approach to show collision orbits have measure zero in general $n$-body systems with standard interactions.

Findings

Collision orbits have measure zero for all energies.

The method applies to gravitational and Coulomb interactions.

Uses symplectic geometry to relate collision volume to Poincaré surface area.

Abstract

For a wide class of two-body interactions, including standard examples like gravitational or Coulomb fields, we show that collision orbits in $n$-body systems are of Liouville measure zero for all energies. We use techniques from symplectic geometry to relate the volume of the union of collision orbits to the area of Poincar\'e surfaces surrounding the collision set.