Dynamics of two interacting particles in classical billiards

TL;DR

This paper investigates the complex dynamics of two interacting particles in a billiard system, revealing how screening influences their behavior and stability, with a focus on a one-dimensional Coulomb interaction.

Contribution

It introduces a coordinate transformation to simplify the analysis of two interacting particles in billiards and explores the effects of screening on their dynamical behavior.

Findings

Screening significantly affects particle dynamics.

Only zero screening length yields bouncing ball behavior.

System exhibits strong non-integrability with resonant islands.

Abstract

The problem of two interacting particles moving in a d-dimensional billiard is considered here. A suitable coordinate transformation leads to the problem of a particle in an unconventional hyperbilliard. A dynamical map can be readily constructed for this general system, which greatly simplifies calculations. As a particular example, we consider two identical particles interacting through a screened Coulomb potential in a one-dimensional billiard. We find that the screening plays an important role in the dynamical behavior of the system and only in the limit of vanishing screening length can the particles be considered as bouncing balls. For more general screening and energy values, the system presents strong non-integrability with resonant islands of stability.