Биллиарды, Ограниченные Дугами Софокусных Квадрик На Плоскости Минковского

Каргинова, Екатерина Евгеньевна; Karginova, E. E.

doi:10.4213/sm9109

Математический сборник

2019

DOI: 10.4213/sm9109

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Биллиарды, Ограниченные Дугами Софокусных Квадрик На Плоскости Минковского

Екатерина Евгеньевна Каргинова¹,

E. E. Karginova

Abstract: Рассмотрены биллиарды в компактных областях на плоскости Минковского, граница которых состоит из дуг софокусных квадрик с углом излома $\le\pi/2$. Получена классификация таких биллиардов, называемых простыми. Описаны первые интегралы и траектории движения шара в простых биллиардах. Подсчитаны инварианты Фоменко-Цишанга для каждого простого биллиарда и доказана теорема о существовании всего трех различных слоений Лиувилля простых биллиардов на плоскости Минковского. Библиография: 23 названия.

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Cited by 8 publications

References 23 publications

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Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

Belozerov¹

2022

Sb. Math.

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We study billiards on compact connected domains in bounded by a finite number of confocal quadrics meeting in dihedral angles equal to . Billiards in such domains are integrable due to having three first integrals in involution inside the domain. We introduce two equivalence relations: combinatorial equivalence of billiard domains determined by the structure of their boundaries, and weak equivalence of the corresponding billiard systems on them. Billiard domains in are classified with respect to combinatorial equivalence, resulting in 35 pairwise nonequivalent classes. For each of these obtained classes, we look for the homeomorphism class of the nonsingular isoenergy 5-manifold, and we show this to be one of three types: either , or , or . We obtain 24 classes of pairwise nonequivalent (with respect to weak equivalence) Liouville foliations of billiards on these domains restricted to a nonsingular energy level. We also define bifurcation atoms of three-dimensional tori corresponding to the arcs of the bifurcation diagram. Bibliography: 59 titles.

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Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

Belozerov¹

2022

Sb. Math.

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Evolutionary force billiards

Фоменко¹,

Ведюшкина²,

Vedyushkina

2022

Известия Российской академии наук. Серия математическая

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Введен новый класс интегрируемых биллиардов, названный эволюционными силовыми биллиардами. Они зависят от параметра и меняют свою топологию с ростом энергии (времени). Доказано, что они реализуют некоторые важные интегрируемые системы с двумя степенями свободы сразу на всем симплектическом четырехмерном фазовом многообразии, а не только на отдельных изоэнергетических $3$-поверхностях. Таковы, например, случай Эйлера и случай Лагранжа. Доказано также, что эти две известные системы "биллиардно эквивалентны", несмотря на то, что первая из них квадратично интегрируема, а вторая допускает линейный интеграл. Библиография: 74 наименования.

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Evolutionary force billiards

Фоменко

Vedyushkina

2022

Izv. Math.

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A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems with two degrees of freedom on the entire symplectic four-dimensional phase manifold at a time, rather than on only individual isoenergy 3-surfaces. For instance, this occurs in the Euler and Lagrange cases. It has also been proved that these two well-known systems are "billiard-equivalent", despite the fact that the former one is square integrable, and the latter one allows a linear integral.

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Биллиарды, Ограниченные Дугами Софокусных Квадрик На Плоскости Минковского

Cited by 8 publications

References 23 publications

Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space

Evolutionary force billiards

Evolutionary force billiards

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