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Ellipsoidal billiards, extremal polynomials, and combinatorics

Hamiltonian Systems

Ellipsoidal billiards, extremal polynomials, and combinatorics
Series: Hamiltonian Systems
Location: ONLINE See abstract
Presenter: Vladimir Dragović, University of Texas at Dallas

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 TdT2d 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 will be via Zoom at:

Passcode: 448487