Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

Abstract

We study billiards on compact connected domains in bounded by a finite number of confocal quadrics meeting in dihedral angles equal to . Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either , or , or . We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.

Billiards and integrable systems

“…For example, given a billiard in a three-dimensional domain Ω ⊂ R 3 bounded by confocal quadrics, links of trajectories lie on straight lines that are simultaneously tangent to a pair of confocal quadrics. Apart from the energy, two other functions are preserved on trajectories [61], [68]. The topological properties of the Liouville foliations for such billiards were presented by Dragović and Radnović [100] and Belozerov [68].…”

“…Apart from the energy, two other functions are preserved on trajectories [61], [68]. The topological properties of the Liouville foliations for such billiards were presented by Dragović and Radnović [100] and Belozerov [68].…”

“…The case of a billiard domain bounded by an ellipsoid in R 3 was considered by Dragović [100]. The classification of confocal domains on a three-dimensional ellipsoid up to rough equivalence was completed by Belozerov [68]. He also considered the topology of Liouville foliations for billiard systems in such domains in the case when a Hooke potential is added (papers in print).…”

“…The problem of the classification of three-dimensional billiards with respect to weak equivalence was solved by Belozerov [68]. For the first time the Liouville foliation for a three-dimensional billiard was described in similar terms by Dragović and Radnović [100] in the case of a three-dimensional billiard inside an ellipsoid.…”

“…The case of billiards on tables of dimension greater than 2 was investigated by Belozerov. He constructed the bifurcation diagrams and classified the table domains bounded by pieces of confocal quadrics in R 3 up to weak equivalence [68]. The topological properties of billiards in such domains in the case when a Hooke potential is added are currently under investigation.…”