ИСТИНА

A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflection from a boundary. You can find overwiew of modern billiard researches in S.L. Tabachnikov’s book. [1] I used Fomenko’s theory [2] to describe topology of joint integrable surfaces. In this report I’m going to discuss billiards bounded by arcs of several confocal conics such that they contains nonconvex angles. Note that such billiards are integrable (the second integral is simply the confocal quadric parameter). Since there is no tangent in vertex of nonvconvex angle, the billiard reflection cannot be defined there in the usual way. The integral trajectory of the flow going into a singular point can not be extend for the all values of the parameter (time). In other words, the corresponding flow is not complete in the sense of the theory of differential equations. In contrast to the classical case of complete flows, the regular leaves of the Liouville foliation are the spheres with handles and punctures, rather than Liouville tori.