Ellipsoidal billiards, extremal polynomials, and partitions
Vladimir Dragović (Univ. Texas at Dallas)
Abstract:
A comprehensive study of periodic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals on the real line. Classification of periodic trajectories is based on a new combinatorial object: billiard partitions.
The case study of trajectories of small periods T, d ≤ T ≤ 2d is given. In particular, it is proven that all d-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates d + 1-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for d = 3.
The talk is based on the following papers:
V. Dragović, M. Radnović, Periodic ellipsoidal billiard trajectories and extremal polynomials, Communications Mathematical Physics, 2019, Vol. 372, p. 183-211.
G. Andrews, V. Dragović, M. Radnović, Combinatorics of the periodic billiards within quadrics, arXiv: 1908.01026, The Ramanujan Journal, DOI: 10.1007/s11139-020-00346-y.
mathematical physicsalgebraic topologycategory theoryquantum algebra
Audience: researchers in the topic
( video )
Topological Quantum Field Theory Club (IST, Lisbon)
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