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This article is cited in 8 scientific papers (total in 8 papers)
Sergey Chaplygin Memorial Issue
Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials
Anani Komla Adabraha, Vladimir Dragovićab, Milena Radnovićcb a Department of Mathematical Sciences, The University of Texas at Dallas,
800 West Campbell Road, 75080 Richardson TX, USA
b Mathematical Institute SANU,
Kneza Mihaila 36, 11001 Beograd, p.p. 367, Serbia
c The University of Sydney, School of Mathematics and Statistics, Carslaw F07, 2006 NSW, Australia
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:
Minkowski plane, relativistic ellipses and hyperbolas, elliptic billiards, periodic and elliptic periodic trajectories, extremal polynomials, Chebyshev polynomials, Akhiezer polynomials, discriminantly separable polynomials.
Received: 02.07.2019 Accepted: 31.08.2019
Citation:
Anani Komla Adabrah, Vladimir Dragović, Milena Radnović, “Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials”, Regul. Chaotic Dyn., 24:5 (2019), 464–501
Linking options:
https://www.mathnet.ru/eng/rcd1022 https://www.mathnet.ru/eng/rcd/v24/i5/p464
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Abstract page: | 147 | References: | 24 |
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