Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials

    2019, Volume 24, Number 5, pp.  464-501

    Author(s): Adabrah A. K., Dragović V., Radnović M.

    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
    Citation: Adabrah A. K., Dragović V., Radnović M., Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials, Regular and Chaotic Dynamics, 2019, Volume 24, Number 5, pp. 464-501

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