Geometry and dynamics I: billiards

Abstract

We consider the simplest possible but nontrivial problem of particle mechanics : on an interval, say [0,1], we consider two particles of the same mass that oscillate. They move about at constant velocity unless there is a collision, either at an endpoint or in encountering each other. The conditions in case of collision are these: when the particle on the left encounters the wall on the left at 0, it rebounds with the same speed and of course in the opposite direction; likewise when the particle on the right rebounds from the wall on the right at the point 1. When they encounter each other (with opposed directions) the condition is what is called an elastic collision, physically one where, after the deformation caused by the collision, the two solids reassume their shapes and retain their combined kinetic energy and momentum. It can be shown that the particles emerge in the opposite directions, but exchange their velocities. The well known and spectacular case is where one is fixed; then the other remains fixed at the point of contact while the first leaves with the same velocity as the particle that hit it. If they encounter each other while going the same direction the result is still the same: the particles exchange their velocities; the particle that was moving faster loses speed to the benefit of the other. The fundamental problem is to describe this simultaneous movement of the two particles over time, but especially when things continue indefinitely or at least for a long period of time. This consideration of infinity is natural in physics when the particles are excited with extremely large velocities and where extremely many collisions occur in a short time interval. In particular, will the motion ultimately be periodic or, to the contrary, will the particles ultimately occupy practically all possible positions?