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

Veduyshkina, Victoria; Кибкало, Владислав Александрович; Pustovoitov, Sergey

doi:10.22405/2226-8383-2021-22-5-44-57

Cited by 5 publications

(1 citation statement)

References 2 publications

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“…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, . .…”

Section: Integrable Generalizations Of Planar Billiards and Billiard ...mentioning

confidence: 99%

Billiards and integrable systems

Fomenko,

Vedyushkina

2023

Russian Math. Surveys

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The survey is devoted to the class of integrable Hamiltonian systems and the class of integrable billiard systems and to the recent results of the authors and their students on the problem of comparison of these classes from the point of view of leafwise homeomorphy of their Liouville foliations. The key tool here are billiards on piecewise planar CW-complexes - topological billiards and billiard books - introduced by Vedyushkina. A construction of the class of evolutionary (force) billiards, introduced recently by Fomenko, is presented, enabling one to model a system in several non-singular energy ranges by a single billiard system, and the use of this class for geodesic flows on two-dimensional surfaces and some systems in mechanics is demonstrated. Some other integrable generalizations of classical billiard systems, including billiards with potentials, billiards in magnetic fields, and billiards with slipping, are discussed. Billiard books with Hooke potentials glued of planar confocal or circular tables, model four-dimensional semilocal singularities of Liouville foliations for integrable systems that contain non-degenerate equilibria. Considering the intersections of several confocal quadrics in $\mathbb{R}^n$ results in a generalization of the Jacobi-Chasles theorem. Bibliography: 144 titles.

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Section: Integrable Generalizations Of Planar Billiards and Billiard ...mentioning

confidence: 99%

Billiards and integrable systems

Fomenko,

Vedyushkina

2023

Russian Math. Surveys

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Topology of Liouville foliations of integrable billiards on table-complexes

Фоменко

Kibkalo²

2022

European Journal of Mathematics

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Billiards and intergrable systems

Fomenko,

Vedyushkina

2023

Uspekhi Mat. Nauk

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Обзор посвящен классу интегрируемых гамильтоновых систем и классу интегрируемых биллиардов, а также недавним результатам авторов и их учеников по задаче сравнения этих классов с точки зрения послойной гомеоморфности их слоений Лиувилля. Ключевым инструментом здесь оказались введенные В. В. Ведюшкиной биллиарды на кусочно-плоских CW-комплексах - топологические биллиарды и биллиардные книжки. Приведено построение класса эволюционных (силовых) биллиардов, введенных недавно А. Т. Фоменко и позволяющих моделировать систему сразу в нескольких неособых зонах энергии при помощи одного биллиарда, а также его применение для геодезических потоков на двумерных поверхностях и систем механики. Обсуждаются другие интегрируемые обобщения классического биллиарда, включая биллиарды с потенциалами, биллиарды в магнитном поле, биллиарды с проскальзыванием. Биллиардные книжки с потенциалом Гука, склеенные из плоских софокусных или круговых столов, моделируют четырехмерные полулокальные особенности слоений интегрируемых систем, содержащие невырожденные положения равновесия. Рассмотрение пересечения нескольких софокусных квадрик в $\mathbb{R}^n$ приводит к обобщению теоремы Якоби-Шаля. Библиография: 144 названия.

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

Cited by 5 publications

References 2 publications

Billiards and integrable systems

Billiards and integrable systems

Topology of Liouville foliations of integrable billiards on table-complexes

Billiards and intergrable systems

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