11-192 Pablo S. Casas and Rafael Ramirez-Ros

Classification of symmetric periodic trajectories in ellipsoidal billiards (1808K, PDF file) Dec 13, 11

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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.

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