Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

In the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars of the Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Lomonosov Moscow State University. 1. Reduction of the degree of integrals for Hamiltonian systems with two degrees of freedom with the help of billiards The integrability of a billiard in a domain bounded by an ellipse was noted by Birkhoff [27]. The integrability of the billiard is preserved if we consider the flat domain bounded by arcs of confocal ellipses and hyperbolas whose boundary does not contain angles equal to 3π 2 . In this case, all the angles of the boundary are equal to π2 , since confocal quadrics always intersect at a right angle. In the book by Kozlov, Treschev [8] it is noted that these dynamical systems are completely Liouville integrable. For the flat billiard in an ellipse, there are coordinates such that the motion is represented as a periodic motion along tori. Up to Liouville equivalence, such systems were studied in detail in [20,21,23] by Dragović, Radnović, and in [28] by Fokicheva. In an interesting paper [22] Dragović and Radnović studied the Liouville foliation for the flat billiard in an ellipse, as well as geodesic flows on the ellipsoid in the Minkowski space, giving an answer in terms of the Fomenko–Zieschang invariants. See also the important papers [24–26] by Dragović and Radnović devoted to the analysis of pseudo-integrable billiards. Fokicheva classified all topological billiards bounded by the arcs of confocal conics (the families of confocal ellipses and hyperbolas and the confocal parabolas) [29,30]. Further, Fokicheva investigated the topology of Liouville foliations on 2010 Mathematics Subject Classification: 37J15; 37J35.