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.
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Funding
The research of V. D. and M. R. was supported by the Discovery Project #DP190101838 Billiards within confocal quadrics and beyond from the Australian Research Council and Project #174020 Geometry and Topology of Manifolds, Classical Mechanics and Integrable Systems of the Serbian Ministry of Education, Technological Development and Science. V. D. would like to thank Sydney Mathematics Research Institute and their International Visitor Program for kind hospitality.
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Adabrah, A.K., Dragović, V. & Radnović, M. Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials. Regul. Chaot. Dyn. 24, 464–501 (2019). https://doi.org/10.1134/S1560354719050034
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DOI: https://doi.org/10.1134/S1560354719050034