Pseudo-Euclidean billiards within confocal curves on the hyperboloid of one sheet

Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology

“…Specific applications include billiards in an ellipsoid in Euclidean and Minkowski spaces. As shown in [37], this technique extends to H. The geometric manifestation of integrability can be seen through the existence of caustics. Namely, all segments of a given billiard trajectory within an H-ellipse are tangent to the same conic, which is confocal with the boundary.…”

“…We assume that the cone is not symmetric, that is, b ̸ = c. Moreover, then without loss of generality we can assume that b < c. Its intersection with H bounds a compact domain on H if and only if all the generatrices of the cone are space-like: see [37]. This happens exactly in one of the following two cases.…”

“…For more details we refer to [37]. We also note the recent paper [50] devoted to geodesics on a hyperboloid in Euclidean space.…”

“…In the present paper we study a new instance of symmetric 2-2-relations, which appears in integrable billiard dynamics in the Minkowski space, on a hyperboloid of one sheet. Such billiards were recently introduced in [37], where Poncelet-type theorems and the corresponding analytic conditions for periodicity were derived. In this work we study periodicity conditions for the dynamics in accordance with the general ideology from [27]: we relate them to extremal polynomials on unions of two intervals.…”

Geodesic scattering on hyperboloids and Knörrer’s map

“…Finally we can mention the problem of geodesic scattering and billiards on quadrics in pseudo-Euclidean case, see [28] and relevant work by Khesin and Tabachnikov [11] and Dragovic, Radnovic and Gasiorek [6,7,9]).…”