Realizing integrable Hamiltonian systems by means of billiard books

Abstract

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.

Billiards and integrable systems

“…For example, using planar and topological billiards with slipping Zav'yalov and these authors [63], [124] succeeded in modelling geodesic flows on the non-orientable surfaces RP 2 and KL 2 . In a magnetic billiard systems it is possible to realize both the molecule A-A with marks r = ∞ and ε = −1 and singularities of complexity 1 splittable in the sense of Zung (see Vedyushkina and Pustovoitov [125], and also [47] and [126]). Billiard books with potentials were used for modelling arbitrary non-degenerate singularities of rank 0.…”

Realization of geodesic flows with a linear first integral by billiards with slipping

“…In parts A and B of this conjecture the problem of realizing atoms-bifurcations and rough molecules was stated; it was solved by Vedyushkina and Kharcheva. Their algorithm for constructing a billiard book that models a prescribed 3-atom or a prescribed rough molecule was described in [18] and [19], respectively; also see [20].…”

Modeling of degenerate peculiarities of integrable billiard systems by billiard books

“…Также подходящими бильярдами реализуются произвольные значения числовых меток [8][9][10] и разнообразные классы гомеоморфности неособых поверхностей постоянной энергии [11,12] и инвариантов Фоменко-Цишанга молекул с числовыми метками [13][14][15]2]. Обзор недавних результатов и открытых задач по топологии интегрируемых бильярдов и гипотезе Фоменко сделан, например, в работах [3,16].…”