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

Kozlov, Valery Vasil'evich

doi:10.1070/rm9707

Cited by 12 publications

(5 citation statements)

References 37 publications

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“…The question of the integrability of billiards in multidimensional domains was also considered by Kozlov [104] and Glutsyuk [105] in recent papers.…”

Section: Integrability Of Billiard Systems and Birkhoff 'S Conjecturementioning

confidence: 98%

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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“…The question of the integrability of billiards in multidimensional domains was also considered by Kozlov [104] and Glutsyuk [105] in recent papers.…”

Section: Integrability Of Billiard Systems and Birkhoff 'S Conjecturementioning

confidence: 98%

Billiards and integrable systems

Fomenko,

Vedyushkina

2023

Russian Math. Surveys

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“…Отметим также, что натуральные системы с потенциалом четвертой степени на плоскости и в n-ом евклидовом пространстве возникают в теории теплоемкости твердых тел. Некоторые вопросы статистической механики и существования полиномиальных интегралов обсуждаются, например, в работах [29,31].…”

Section: случай невырожденной матрицы Bunclassified

Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups

Бизяев¹,

Borisov²,

Kilin³

et al. 2017

Nelin. Dinam.

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“…Theorem 2.4 implies much weaker results, but it needs weaker assumptions. For example, it does not need compactness of the configuration space, so it can be used in situations studied in [17].…”

Section: Billiards With Thin Scatterersmentioning

confidence: 99%

Degenerate billiards

Болотин

2016

Proc. Steklov Inst. Math.

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In an ordinary billiard system trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Then collisions are rare. We study trajectories of degenerate billiards which have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an antiintegrable limit.

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

Cited by 12 publications

References 37 publications

Billiards and integrable systems

Billiards and integrable systems

Integrability and nonintegrability of sub-Riemannian geodesic flows on Carnot groups

Degenerate billiards

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