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

Vedyushkina, V. V.; Kharcheva, I. S.

doi:10.1070/sm9468

Cited by 13 publications

(3 citation statements)

References 14 publications

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“…Conjecture B has also been proved [40]. The proof uses billiard tables B 0 disjoint from the focal line and bounded by two arcs if ellipses and two arcs of hyperbolae.…”

Section: Conjecture Fomenko the Following Objects Can Be Realized In ...mentioning

confidence: 93%

“…Several theses in Fomenko's conjecture have already been proved: it was shown that one can realize all non-degenerate 3-atoms [32], [36], all values of numerical marks [37]- [39], as well as an arbitrary Fomenko invariant without marks [40]. In other words, no 'component' of the invariant can alone be an obstruction to realization, and moreover, billiards model all classes of relatively weaker (namely, rough Liouville) equivalence.…”

Section: Introductionmentioning

confidence: 95%

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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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“…Conjecture B has also been proved [40]. The proof uses billiard tables B 0 disjoint from the focal line and bounded by two arcs if ellipses and two arcs of hyperbolae.…”

Section: Conjecture Fomenko the Following Objects Can Be Realized In ...mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 95%

Billiards and integrable systems

Fomenko,

Vedyushkina

2023

Russian Math. Surveys

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“…Considerable advancements have also been made in the proof of Fomenko's general conjecture on billiards which was stated in [19]: Vedyushkina and Kharcheva showed that using billiard books one can realize arbitrary Bott 3-atoms [4], [5] and the base of a Liouville foliation with these singularities [20], so that any molecule without numerical marks can be realized. Fomenko, Vedyushkina and Kibkalo investigated the 'local' version of the general conjecture (see [21]): they showed that arbitrary values of the marks r and ε on an edge of a molecule (see [22]) and the integer mark n on a family subgraph (see [23]) can be realized, as well as certain combinations of marks (see [24]).…”

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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Topology of Liouville foliations of integrable billiards on table-complexes

Фоменко

Kibkalo²

2022

European Journal of Mathematics

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

Cited by 13 publications

References 14 publications

Billiards and integrable systems

Billiards and integrable systems

Billiard with slipping by an arbitrary rational angle

Topology of Liouville foliations of integrable billiards on table-complexes

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