Bouncing ball orbits and symmetry breaking effects in a three-dimensional chaotic billiard

We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90 degrees, and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observe a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for un equal radii.Comment: 11 pages, 10 eps figure

Geometric Hermite Interpolation with Maximal Order and Smoothness

this paper we prove the conjecture in the simplest case, for planar quadratic spline curves. In addition to an analysis of the GHI in Sections 3 and 4 we give in Section 2 a simple construction principle for curvature continuous quadratic splines. Some examples are provided in Section 5. We hope that the particularly elegant solutions possible in this simple case will stimulate further research on our conjecture. At least the cubic case seems to be within reach, both for planar and for space curves. Of course, from a numerical point of view, the implementation of a general GHI scheme is straightforward. All that is required is a routine for solving polynomial systems of moderate size. 2 Control Polygons As is shown in Figure 3, the B'ezier points of a continuously differentiable quadratic spline curve are obtained simply by subdividing the edges of its control polygon. More precisely, the B'ezier points of the j-th segment ar