Classification of symmetric periodic trajectories in ellipsoidal billiards
Dynamical Systems
2015-05-04 v1 Exactly Solvable and Integrable Systems
Abstract
We classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of R^{n+1} without any symmetry of revolution. 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
Cite
@article{arxiv.1203.6188,
title = {Classification of symmetric periodic trajectories in ellipsoidal billiards},
author = {Pablo S. Casas and Rafael Ramírez-Ros},
journal= {arXiv preprint arXiv:1203.6188},
year = {2015}
}
Comments
26 pages, 77 figures, 17 tables