The Fomenko-Zieschang invariants of nonconvex topological billiards

Vedyushkina, V. V.

doi:10.1070/sm9041

Cited by 27 publications

(11 citation statements)

References 21 publications

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“…Recently, a new class of integrable topological billiards has been discovered. They are realized as the dynamics of a material point on the two-dimensional locally-Euclidean cell complexes, the edges of which are arcs of confocal quadrics (see [6]). The corresponding Hamiltonian system is realized on a four-dimensional piecewise-smooth manifold and (after reduction) on a three-dimensional piecewise-smooth isoenergy surface.…”

Section: Reduction Of the Degree Of Integrals For Hamiltonian Systemsmentioning

confidence: 99%

“…Let us consider the 2-dimensional cell complex obtained by gluing the elementary billiards along common segments of their boundaries. In case when along each edge no more than two billiards-sheets are glued together, the resulting manifold is called a topological billiard (see [6]). In the case when along some edge more than two billiards are glued, this edge must be endowed by a permutation ∈ , where is the number of billiard sheets glued along this edge.…”

Section: Reduction Of the Degree Of Integrals For Hamiltonian Systemsmentioning

confidence: 99%

“…The complete classification of the topological billiards up to Liouville equivalence was done by Vedyushkina in [6]. In the papers [9, 10] Fomenko, Kharcheva, Vedyushkina investigated the topology of the Liouville foliations for the billiard books.…”

Section: Reduction Of the Degree Of Integrals For Hamiltonian Systemsmentioning

confidence: 99%

See 2 more Smart Citations

Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

Фоменко

Vedyushkina

2019

Theor appl mech (Belgr)

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Section: Reduction Of the Degree Of Integrals For Hamiltonian Systemsmentioning

confidence: 99%

Section: Reduction Of the Degree Of Integrals For Hamiltonian Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

Фоменко

Vedyushkina

2019

Theor appl mech (Belgr)

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“…The class of topological billiards was fully classified by Vedyushkina [31], [33], both structurally (as CW-complexes) and topologically (that is, the Fomenko-Zieschang invariants were calculated). A structure equivalence between tables was defined, which preserves the topology of the Liouville foliation and admits a continuous deformation of boundary arcs in the class of confocal quadrics.…”

Section: Introductionmentioning

confidence: 99%

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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“…Such systems were then realized using several billiards (see [4][5][6]): for a regular energy zone of a system (for any of its Q 3 ) its own billiard with the same Fomenko-Zieschang invariant was constructed. For example, for geodesic flows of Riemannian metrics (e.g., on conics [7]) and for billiards introduced by Vedyushkina on CW complexes (billiard books [8] and topological billiards [9]), the particle's energy is only a scaling parameter. It turns out that these classes are important in applications.…”

mentioning

confidence: 99%