TITLE:
The classification of symmetric periodic trajectories in ellipsodal billiards

AUTHORS:
Pablo S. Casas, and Rafael Ramirez-Ros

Departament de Matematica Aplicada I
Universitat Politecnica de Catalunya
Diagonal 647, 08028 Barcelona, Spain

E-MAIL ADDRESSES:
pablo@casas.upc.edu, rafael.ramirez@upc.edu

URLs:
www.ma1.upc.edu/~casas/
www.ma1.upc.edu/~rafael/

ABSTRACT:
We find and classify nonsingular symmetric periodic trajectories (SPTs)
of billiards inside nondegenerate ellipsoids of $R^{n+1}$. SPTs are defined
as periodic trajectories passing through some symmetry set. We prove that
there are exactly $2^{2n}(2^{n+1}−1)$ classes of such trajectories.
We have implemented an algorithm to find minimal SPTs of each of the 12
classes in the 2D case ($R^2$) and each of the 112 classes in the 3D
case ($R^3$). They have periods 3, 4 or 6 in the 2D case; and 4, 5,
6, 8 or 10 in the 3D case. We display a selection of 3D minimal SPTs.
Some of them have properties that cannot take place in the 2D case.

KEYWORDS:
Billiards, integrability, reversors, periodic orbits