Force Evolutionary Billiards and Billiard Equivalence of the Euler and Lagrange Cases

Vedyushkina, V. V.; Фоменко, А. Т.

doi:10.1134/s1064562421010154

Cited by 11 publications

(4 citation statements)

References 14 publications

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“…The construction of an evolutionary (or force) billiard was proposed by Fomenko [57]. It enables one to model a system to both sides of a singular value of h without modelling the foliation on the singular surface (which usually contains an equilibrium or a degenerate singularity).…”

Section: Evolutionary Billiards and Billiard Equivalence Of Integrabl...mentioning

confidence: 99%

“…Theorem 17 ([57], [58]). On each regular symplectic 4-sheet M 4 g the Lagrange integrable case can be realized (in the sense of Liouville equivalence) by one of the five force billiards constructed in [57]. Their force billiards are bounded by concentric circles (and therefore integrable at each moment of their evolution).…”

Section: 2mentioning

confidence: 99%

“…1) construct a billiard modelling a system in question in several distinct energy ranges (perhaps leaving out several energy levels of the system); 2) model topologically, by means of billiards 4-dimensional singularities of integrable systems, in particular, non-degenerate semilocal singularities (of centre-centre, centre-saddle, saddle-saddle, focus-focus type), and degenerate singularities common in applications (saddle-node, elliptic and hyperbolic forks, period doubling: see [56]). A new approach to the solution of the first problem was proposed by Fomenko: he introduced [57] the class of evolutionary (or force) billiard systems, in which the geometry of the billiard table and the reflection law depend on a parameter and can have bifurcations for some values of the parameter. Non-singular ranges of the parameter correspond to billiard-book levels that can model the system in non-singular energy ranges (so that the Fomenko-Zieschang invariants, that is, classes of leafwise homeomorphy, are the same).…”

Section: Introductionmentioning

confidence: 99%

“…In § 3 we describe in detail the class of evolutionary billiards introduced recently [57], [58] and present result on modelling, by means of such systems, the topology of the (whole) Euler and Largange tops (so that all non-singular foliations on the symplectic phase space M 4 of the system are realized by a single evolutionary billiard), the Joukowsky system, and the Kovalevskaya top [59].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Evolutionary Billiards and Billiard Equivalence Of Integrabl...mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Billiards and integrable systems

Fomenko,

Vedyushkina

2023

Russian Math. Surveys

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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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Billiards with Changing Geometry and Their Connection with the Implementation of the Zhukovsky and Kovalevskaya Cases

Фоменко

Vedyushkina

2021

Russ. J. Math. Phys.

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Force Evolutionary Billiards and Billiard Equivalence of the Euler and Lagrange Cases

Cited by 11 publications

References 14 publications

Billiards and integrable systems

Billiards and integrable systems

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

Billiards with Changing Geometry and Their Connection with the Implementation of the Zhukovsky and Kovalevskaya Cases

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