Quadrics in Pseudo-Euclidean Spaces, Integrable Billiards and Extremal Polynomials

We study the geometry of confocal quadrics in pseudo-Euclidean spaces of dimensions 2, 3, and 4, respectively. Along with the notion of geometric quadrics, we also investigate the relativistic quadrics which provide tools for further investigations of billiard dynamics. The geometric quadrics of a confocal pencil and their types in pseudo-Euclidean spaces do not share all of the usual properties with confocal quadrics in Euclidean spaces, including those necessary for applications in billiard dynamics and separable mechanical systems in general. For instance, in n-dimensional Euclidean space, there are n geometric types of quadrics, whereas in n-dimensional pseudo-Euclidean space, there are n + 1 geometric types of quadrics. Relativistic quadrics enable us to define and use Jacobi coordinates in pseudoEuclidean settings. In the study of periodic billiard trajectories, we distinguish two cases: trajectories which are periodic with respect to Cartesian coordinates, which are the usual periodic trajectories, and the so-called elliptic periodic trajectories, which are periodic with respect to Jacobi coordinates. In the Minkowski plane, we derive necessary and sufficient conditions for periodic and elliptic periodic trajectories of billiards within an ellipse in terms of an underlying elliptic curve. We derive equivalent conditions in terms of polynomial equations as well. The corresponding polynomials are related to the classical extremal polynomials. We have indicated the similarities and differences with respect to previously studied periodic billiard trajectories in Euclidean cases. The classification of hypersurfaces of degree 2 in four-dimensional pseudo-Euclidean space has been done in signatures (3, 1) and (2, 2).