Pseudo-integrable billiards and double reflection nets

Dragović, Vladimir; Раднович, Милена; Radnović, Milena

doi:10.4213/rm9648

Uspekhi Mat. Nauk

2015

DOI: 10.4213/rm9648

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Pseudo-integrable billiards and double reflection nets

Vladimir Dragović¹,

Милена Раднович²,

Milena Radnović³

Abstract: Дается обзор двух недавно возникших направлений в исследовании биллиардов в пучках квадрик: псевдоинтегрируемые биллиарды на плоскости и решетки двойных отражений. Библиография: 48 названий. Ключевые слова: эллиптический биллиард, многоугольный биллиард, периодические траектории, конфигурация двойного отражения, дискретные интегрируемые системы, конгруэнции прямых, псевдоинтегрируемость.

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Cited by 7 publications

References 38 publications

(51 reference statements)

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Two-Valued Groups, Kummer Varieties, and Integrable Billiards

Buchstaber

Dragović

2018

Arnold Math J.

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A natural and important question of study two-valued groups associated with hyperelliptic Jacobians and their relationship with integrable systems is motivated by seminal examples of relationship between algebraic two-valued groups related to elliptic curves and integrable systems such as elliptic billiards and celebrated Kowalevski top. The present paper is devoted to the case of genus 2, to the investigation of algebraic two-valued group structures on Kummer varieties.One of our approaches is based on the theory of σ-functions. It enables us to study the dependence of parameters of the curves, including rational limits. Following this line, we are introducing a notion of n-groupoid as natural multivalued analogue of the notion of topological groupoid.Our second approach is geometric. It is based on a geometric approach to addition laws on hyperelliptic Jacobians and on a recent notion of billiard algebra. Especially important is connection with integrable billiard systems within confocal quadrics.The third approach is based on the realization of the Kummer variety in the framework of moduli of semi-stable bundles, after Narasimhan and Ramanan. This construction of the two-valued structure is remarkably similar to the historically first example of topological formal two-valued group from 1971, with a significant difference: the resulting bundles in the 1971 case were "virtual", while in the present case the resulting bundles are effectively realizable.MSC2010: 20N20, 14H40, 14H70

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Two-Valued Groups, Kummer Varieties, and Integrable Billiards

Buchstaber

Dragović

2018

Arnold Math J.

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Polynomial conservation laws for the Lorentz gas and the Boltzmann-Gibbs gas

Kozlov¹

2016

Russ. Math. Surv.

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Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems

Vedyushkina¹,

Kharcheva²

2021

Sb. Math.

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Pseudo-integrable billiards and double reflection nets

Cited by 7 publications

References 38 publications

Two-Valued Groups, Kummer Varieties, and Integrable Billiards

Two-Valued Groups, Kummer Varieties, and Integrable Billiards

Polynomial conservation laws for the Lorentz gas and the Boltzmann-Gibbs gas

Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems

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