The general scenario

Elliptic billiards is the name given to a particular class of discrete time (DT) mechanical system in which a particle moves in continuous time (CT) and space but is observed only at a discrete set of times whenever it bounces off a fixed surface. It is therefore a form of stroboscopic mechanics but, unlike conventional stroboscopic mechanics where the time intervals between successive observations are determined by the observer and are usually of equal duration, the time between successive observations in elliptic billiards is variable, being determined by the dynamical behaviour of the system under observation. We shall first discuss this form of mechanics using a purely geometrical approach (Moser and Veselov, 1991) and then we shall show that the same system can be analysed in terms of Lee mechanics, which was studied in the previous chapter.

We will discuss the situation when the particle under observation is confined to the interior of some container such as an ellipsoid (hence the title of this chapter), being observed only when it bounces elastically off the surface of that container, Figure 11.1. The container can in principle be any closed shape, i.e., not necessarily a box with rectangular sides. In the scenario we are interested in in this chapter, the particle is assumed to move inertially, i.e., it moves uniformly between successive impacts on the surface of the container.