ИСТИНА

Let us consider a standard billiard problem in some fixed domain. A particle moves in a straight line within the domain. When the particle hits the boundary it reflects from it without velocity loss. This dynamic system contains the first integral - the scalar square of the velocity vector. In some special cases such a system has another integral. One of these cases is a billiard in the domain bounded by confocal quadrics. The second special integral can be described by the following feature of trajectories: the straight lines containing the segments of the polygonal billiard trajectory are tangents to a certain quadric (ellipse or hyperbola). This dynamical system has 4-dimensional phase space and two integrals. One of them is a Hamiltonian. Integrable Hamiltonian systems with 2 degrees of freedom have Fomenko-Zieschang invariants. Such invariants allow us to speak about the equivalence between closures of trajectories - Liouville equivalence. Up to Liouville equivalence, such systems have been studied in details in by V. Dragovich, M. Radnovich, and by V.V. Fokicheva. A billiard's book is a generalization of these billiards. Such type of billiards formed by gluing a few classical billiard domains along pieces of their boundaries. The special case where we glue two domains called a topological billiard and was researched by V.V. Fokicheva. Researching billiard's books we try model famous integrable systems in terms of Fomenko-Zieschang invariants. The Fomenko conjecture about modeling Fomenko-Zieschang invariants using billiard's books and new results that confirm the part of the conjecture will be presented.