Realizing integrable Hamiltonian systems by means of billiard books

Fomenko’s conjecture that the topology of the Liouville foliations associated with integrable smooth or analytic Hamiltonian systems can be realized by means of integrable billiard systems is discussed. An algorithm of Vedyushkina and Kharcheva’s realizing 3-atoms by billiard books, which has been simplified significantly by formulating it in terms of f f -graphs, is presented. Note that, using another algorithm, Vedyushkina and Kharcheva have also realized an arbitrary type of the base of the Liouville foliation on the whole 3-dimensional isoenergy surface. This algorithm is illustrated graphically by an example where the invariant of the well-known Joukowsky system (the Euler case with a gyrostat) is realized for a certain energy range. It turns out that the entire Liouville foliation, rather than just the class of its base, is realized there; that is, the billiard and mechanical systems turn out to be Liouville equivalent. Results due to Vedyushkina and Kibkalo on constructing billiards with arbitrary values of numerical invariants are also presented. For billiard books without potential that possess a certain property, the existence of a Fomenko–Zieschang invariant is shown; it is also proved that they belong to the class of topologically stable systems. Finally, an example is presented when the addition of a Hooke potential to a planar billiard produces a splitting nondegenerate 4-singularity of rank 1.