Three unequal masses on a ring and soft triangular billiards

TL;DR

This paper explores the dynamics of three soft-interacting particles on a ring, demonstrating their correspondence to a soft triangular billiard and analyzing how mass and softness ratios influence the system.

Contribution

It introduces a novel mapping between three soft particles on a ring and a soft triangular billiard, highlighting the dependence on mass and softness ratios.

Findings

Dynamics depend only on mass and softness ratios.

Transition from soft to hard interactions can be studied via specific potentials.

Numerical examples demonstrate the theoretical correspondence.

Abstract

The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the {\it masses ratio} between particles and {\it softness ratio} of the particles interaction. The transition from soft to hard interaction can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.