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

Vedyushkina, V. V.; Zav'yalov, Vladimir Nikolaevich

doi:10.4213/sm9772e

Cited by 5 publications

(3 citation statements)

References 47 publications

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“…The new classes of billiards have successfully been used in modelling of complex 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]).…”

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

confidence: 99%

“…Theorem 25 (Vedyushkina and Zav'yalov [124]). A geodesic flow on a nonorientable two-dimensional manifold (a Klein bottle or a projective plane) that has an additional first integral which is linear in momenta is Liouville equivalent to a billiard with slipping formed by planar billiards bounded by concentric circles.…”

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

confidence: 99%

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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%

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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“…Circular billiards with slipping by an angle of π are integrable and enable one to realise an arbitrary geodesic flow on the projective plane RP 2 or the Klein bottle Kl 2 that is integrable by a linear first integral (see [30]). In a similar way slipping by an angle of π along elliptic boundaries of confocal billiards was defined in [28].…”

Section: § 1 Introductionmentioning

confidence: 99%

Billiard with slipping by an arbitrary rational angle

Zav'yalov

2023

Sb. Math.

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The class of billiards in a disc with slipping along the boundary circle by an angle commensurable with $\pi$ is considered. For such billiards it is shown that an isoenergy surface of the system is homeomorphic to a lens space $L(q,p)$ with parameters satisfying $0 < p <q$. The set of pairs $(q, p)$ such that there exists a billiard in a disc realizing the corresponding lens space $L(q,p)$ is described in terms of solutions of a linear Diophantine equation in two variables. This result also holds for planar billiards with slipping in simply connected domains with smooth boundary, that is, it is not confined to the integrable case. Bibliography: 30 titles.

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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 geodesic flows with a linear first integral by billiards with slipping

Cited by 5 publications

References 47 publications

Billiards and integrable systems

Billiards and integrable systems

Billiard with slipping by an arbitrary rational angle

Billiards and intergrable systems

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