|Authors:||Adabrah, Anani Komla
|Title:||Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials||Journal:||Regular and Chaotic Dynamics||Volume:||24||Issue:||5||First page:||464||Last page:||501||Issue Date:||1-Sep-2019||Rank:||M21||ISSN:||1560-3547||DOI:||10.1134/S1560354719050034||Abstract:||
We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe a relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. Similarities and differences with respect to the previously studied Euclidean case are highlighted.
|Keywords:||14H70 | 26C05 | 37J35 | 41A10 | 70H06 | Akhiezer polynomials | Chebyshev polynomials | discriminantly separable polynomials | elliptic billiards | extremal polynomials | Minkowski plane | periodic and elliptic periodic trajectories | relativistic ellipses and hyperbolas||Project:||Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems
Australian Research Council, Discovery Project #DP190101838
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