Next: Appendix: Is there a Up: On nonconvex caustics of Previous: Are there examples of


Caustics and billiards in three dimensions

A three-dimensional Birkhoff billiard is defined by a differentiable convex surface R in tex2html_wrap_inline1344 . The return map to the boundary defines a map tex2html_wrap_inline894 on a four-dimensional manifold tex2html_wrap_inline1552 : two successive impact points tex2html_wrap_inline1554 define the next pair tex2html_wrap_inline1556 . Caustics for such billiards do not exist [3] [12]. One can consider however differential geometric caustics of three-dimensional billiards which are defined through the wave front (see Figure 16). We now illustrate this in the case of a surface of equal thickness. Take a curve T of constant width in the x-z plane in tex2html_wrap_inline1344 . Assume that T is reflection symmetric with respect to the z-axes L. If we rotate T around L, we obtain a surface R of revolution which is at the same time a surface of constant width (see [6]). A billiard trajectory orthogonal to a point P of R intersects the surface R a second time orthogonally. It corresponds to a 2-periodic orbit of the return map tex2html_wrap_inline894 . By symmetry, a two-dimensional differential geometrical caustic of a three-dimensional billiard is obtained by rotating the caustic tex2html_wrap_inline880 of T around L.

tex2html_wrap1616 tex2html_wrap1618
tex2html_wrap1620 tex2html_wrap1622
The symmetry of the surface of revolution leads to a billiard map tex2html_wrap_inline894 which has an angular momentum integral I. One has to understand the dynamics of tex2html_wrap_inline894 on three-dimensional leaves I=const. When the angular momentum is zero, an orbit moves in a plane through the rotation axis L. The three-dimensional zero angular momentum leaf is foliated by two-dimensional invariant manifolds on which the dynamics is the two-dimensional billiard map (see Figure 14).

tex2html_wrap1624 tex2html_wrap1626
Having a two-dimensional Birkhoff billiard as a subsystem of the three- dimensional billiard implies the existence of many periodic orbits, more than are known to exist in general [2]. Also, by the variational theorem in ergodic theory, the topological entropy of the three-dimensional billiard map is bounded below by the topological entropy of the two-dimensional billiard. It would be interesting to know more about the dynamics of tex2html_wrap_inline894 on a nonzero angular momentum leaf. Figure 15 shows some typical numerically computed trajectories.

tex2html_wrap1628 tex2html_wrap1630
tex2html_wrap1632 tex2html_wrap1634
tex2html_wrap1636 tex2html_wrap1638
tex2html_wrap1640 tex2html_wrap1642
tex2html_wrap1644

Figure 16. The standard Legendre collapse of a wave front moving from a surface of revolution of constant width. Two-dimensional sections of this collapse are shown in Figure 11a. The surface turns inside out during this collapse as can be seen by the arrows which point inside the surface at the beginning and outside at the end. Differing from the famous movie, this "turning the sphere inside out" is not smooth: the smoothness fails on points of the caustic.
Movie (250 K Gif movie) of the Standard collapse.
Next: Appendix: Is there a Up: On nonconvex caustics of Previous: Are there examples of

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