Integrable Mechanical Billiards in Higher-Dimensional Space Forms

    2024, Volume 29, Number 3, pp.  405-434

    Author(s): Takeuchi A., Zhao L.

    In this article, we consider mechanical billiard systems defined with Lagrange's integrable {extension} of Euler's two-center problems in the Euclidean space, the sphere, and the hyperbolic space of arbitrary dimension $n \geqslant 3$. In the three-dimensional Euclidean space, we show that the billiard systems with any finite combinations of {spheroids and circular hyperboloids of two sheets} having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and in the three-dimensional hyperbolic space by means of central projection. Using the same approach, we also extend these results to the $n$-dimensional cases.
    Keywords: mechanical billiard systems, Euler's two-center problem, Lagrange problem, integrability
    Citation: Takeuchi A., Zhao L., Integrable Mechanical Billiards in Higher-Dimensional Space Forms, Regular and Chaotic Dynamics, 2024, Volume 29, Number 3, pp. 405-434

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