A billiard-theoretic approach to elementary one-dimensional elastic collisions

Abstract

A simple relation is developed between elastic collisions of freely-moving point particles in one dimension and a corresponding billiard system. For two particles with masses m1 and m2 on the half-line x > 0 that approach an elastic barrier at x = 0, the corresponding billiard system is an infinite wedge. The collision history of the two particles can be easily inferred from the corresponding billiard trajectory. This connection nicely explains the classic demonstrations of the "dime on the superball" and the "baseball on the basketball" that are a staple in elementary physics courses. It is also shown that three elastic particles on an infinite line and three particles on a finite ring correspond, respectively, to the motion of a billiard ball in an infinite wedge and on on a triangular billiard table. It is shown how to determine the angles of these two sets in terms of the particle masses.

Simplest piston problem. I. Elastic collisions

“…Consequently, a naive analysis of successive billiard collisions becomes θ θ 000 000 000 000 000 000 000 000 000 000 prohibitively cumbersome. However, a considerable simplification is achieved by introducing the "billiard" coordinates [10,11,12,13,14] …”

“…4, the allowed region for the billiard is the interior of a highly skewed tetrahedron whose two acute interior angles are given by θ = tan −1 1/m 2 . While this geometry may seem complicated at first sight, these coordinates ensure that all billiard collisions with domain boundaries are specular [12,13,14], and this feature greatly simplifies the problem.…”

“…Then in Sec. III, we map the trajectories of three particles on the line onto an equivalent elastic billiard particle that moves within a highly skewed tetrahedron, with the specular reflection whenever the billiard hits the tetrahedron boundaries [10,11,12,13,14]. From this simple geometrical mapping, we can understand many of the unusual dynamical properties of the system, as will be discussed in Sec IV.…”

The Bouncing Penny and Nonholonomic Impacts

“…This is characterized by a particle on a table moving in straight lines and reflecting off of the boundary by specular reflection. Some examples where this is studied is [2], [3], [4], [5], [6] and §9.2 of [7]. Moreover, this problem has even been applied to biological processes [8].…”

Creating entanglement using integrals of motion

“…We show that an initial disentangled state can evolve into one where the heavy and light particles are entangled, and propose a sensor, containing N total atoms, with a √ N total times higher sensitivity than in a one-atom sensor with N total repetitions. In a 1D system of hard-core particles, by tuning the ratios between the particle masses one can choose a variety of distinct regimes of motion [1,2]. While generic values result in thermalization, for some special ones the system maps to known multidimensional kaleidoscopes [3,4].…”