Next: Caustics of convex billiards Up: On nonconvex caustics of Previous: Introduction


Convex billiard tables

Good starting points in the literature of the billiard problem are [23] [4] [24] or [21]. In the case, when the domain is a strictly convex planar region, the return map to the boundary T defines a map tex2html_wrap_inline894 of the annulus tex2html_wrap_inline896 . The circle tex2html_wrap_inline898 parametrizes the table with impact point s at the boundary T. The angle of impact tex2html_wrap_inline864 and s determine the next reflection point tex2html_wrap_inline908 with angle tex2html_wrap_inline910 . The reader can see some orbits of the map tex2html_wrap_inline912 in Figures 3 and 4.

Noncontractible invariant curves tex2html_wrap_inline878 in A can define caustics tex2html_wrap_inline880 in the interior of the table T (see Figure 2). Caustics are curves with the property that a ball, once tangent to tex2html_wrap_inline880 stays tangent after every reflection. We will discuss the precise relation between the invariant curve tex2html_wrap_inline878 and the caustic tex2html_wrap_inline880 in the next section. Examples of caustics are drawn in Figures 6,7 and Figure 11a.

tex2html_wrap980 tex2html_wrap982
tex2html_wrap984 tex2html_wrap986
Invariant curves tex2html_wrap_inline878 of tex2html_wrap_inline894 are interesting for the dynamics because they divide the phase space A into invariant regions and therefore prevent ergodicity. They often occur near the boundary of a smooth, strictly convex table and define 'Lazutkin whisper galleries` [8]. Most convex billiards have no invariant curves [11]. Also, if the curvature of the table is unbounded at some point, then there are no invariant curves in a neighborhood of the boundary of A [15]. Strict convexity is necessary for the existence of caustics [17]. The billiard in Figure 1 for example has no caustics.

tex2html_wrap988 tex2html_wrap990 tex2html_wrap992
tex2html_wrap994 tex2html_wrap996 tex2html_wrap998
tex2html_wrap1000 tex2html_wrap1002 tex2html_wrap1004
tex2html_wrap1006 tex2html_wrap1008 tex2html_wrap1010
We will focus in this article especially on billiard tables of constant width. They all leave the curve tex2html_wrap_inline970 invariant. Figures 3 and 4 show typical trajectories in such tables. The flatness of the invariant curve tex2html_wrap_inline878 will allow us to make statements about the regularity of the corresponding caustics tex2html_wrap_inline880 depending on the regularity of the table. First however, in the next section, we will clarify the relation between the invariant curve tex2html_wrap_inline878 in the phase space and the caustic tex2html_wrap_inline880 in the interior of the table.

Next: Caustics of convex billiards Up: On nonconvex caustics of Previous: Introduction

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