Realization of focal singularities of integrable systems using billiard books with a Hooke potential field

Abstract

Рассмотрены системы движения частицы в поле центрального потенциала Гука по биллиардной книжке, склеенной из плоских круговых биллиардов. Важный класс невырожденных фокусных особенностей ранга 0 интегрируемых систем с 2 степенями свободы полностью реализован таким классом биллиардов. А именно, для каждой полулокальной фокусной особенности была построена биллиардная система с особенностью, послойно гомеоморфной данной.Ключевые слова: интегрируемый биллиард, фокусная особенность, слоение Лиувилля.

Billiards and integrable systems

“…For singularities of centre-centre, centre-saddle, and saddle-saddle type the approach proposed and developed by Kibkalo [65], [66] uses confocal billiards (as limits of geodesic flows on an ellipsoid) and sets of permutations found by Oshemkov for saddle singularities of smooth systems. Focus-focus singularities were modelled by Vedyushkina, Kibkalo, and Pustovoitov [67] by means of billiard books with attracting Hooke potentials which are glued of n discs with permutation (1, . .…”

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

“…For instance, the class of evolutionary force billiards introduced by Fomenko (when the geometry of the table varies with the energy of the billiard ball) enables one to model integrable systems in several nonsingular energy ranges simultaneously: see [31]- [33]. On the other hand, adding a Hooke potential to a topological billiard or a billiard book has made it possible to advance significantly in the problem of modelling nondegenerate local and semilocal singularities of rank 0, that is, neighbourhoods of an equilibrium of a system and the leaf of the foliation containing it, by means of billiards [34]- [36]. § 2.…”