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Introduction

The game of billiards in a general convex table is named after G.D. Birkhoff , who proposed the problem in 1927. Birkhoff used it as an illuminating example to illustrate results obtained in connection with the Newtonian three body problem. Birkhoff billiards are examples of dynamical systems, where results in analysis, calculus of variations or ergodic theory find applications. The problem is closely related to the Dirichlet problem on the domain enclosed by the table. Birkhoff viewed it also as a limiting case for the geodesic flows on two-dimensional surfaces.

In contrast from the real billiard game, where relatively short orbits are considered and other factors like the spin of the ball are relevant, the more academic Birkhoff billiard deals with trajectories of arbitrary length, hitting the boundary according to the precise law of reflection. An orbit is determined by the starting point T(s) of the table and the angle tex2html_wrap_inline864 between the velocity direction and the tangent at the point T(s).

If we shoot a billiard ball and follow the individual angles of impact, it is natural to ask what is the minimal or maximal angle which occurs along an orbit. Is it possible to shoot in such a way that a ball hits the boundary in the future with angles arbitrarily close to 0 and at other times with angles arbitrarily close to tex2html_wrap_inline870 ?

tex2html_wrap888 tex2html_wrap890

Birkhoff realized that such an orbit does not exist if and only if there is a continuous invariant curve tex2html_wrap_inline874 in the phase space of all pairs tex2html_wrap_inline876 . Such an invariant curve tex2html_wrap_inline878 in the phase space is associated to a caustic tex2html_wrap_inline880 in the interior of the table. The curve tex2html_wrap_inline880 has the property that if the path of a ball has a tangency at tex2html_wrap_inline880 , then also the reflected path has a tangency at tex2html_wrap_inline880 .

An aim of our article is to look closer at the geometry of caustics and to illustrate the story with some pictures. By relating the caustics of a class of billiards with differential geometrical caustics, we will see that this subject is of more general interest. Differential geometric caustics are relevant in applied sciences including observational astronomy and earthquake science: assume that a light source is switched on in a nonhomogeneous or nonflat medium or assume that an earthquake starts at its epicenter. The set of points which are reached by the light or the sound wave after a given time is called a wave front. For small times, such a wave front is diffeomorphic to a sphere. If the medium is not homogeneous or not flat, these spheres become more and more deformed as time increases and there will be in general points where the front ceases to be an immersed surface, for example points where different parts of the front selfintersect. The set of all points, where the immersion fails, is called the caustic of the light or acoustic source. As we will illustrate, there can be relations between differential geometric caustics and caustics in billiards.

Our story, which begins in the next section can be read first in the fast lane by following the text to the figures.

Next: Convex billiard tables Up: On nonconvex caustics of Previous: On nonconvex caustics of

Oliver Knill
Fri Jun 12 13:34:37 CDT 1998