Saddle Singularities in Integrable Hamiltonian Systems: Examples and Algorithms

Фоменко, А. Т.; Kibkalo, Vladislav

doi:10.1007/978-3-030-50302-4_1

Cited by 12 publications

(6 citation statements)

References 34 publications

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“…A long list of publications on this subject was presented in books by Fomenko [78] and Bolsinov and Fomenko [6], and in papers by Fomenko and Vedyushkina [55], [79], and Fomenko and Kibkalo [34], [66].…”

Section: )mentioning

confidence: 99%

“…The question, raised by Fomenko and Vedyushkina [35], on modelling nondegenerate singularities of rank 0 (that is, of foliations in a neighbourhood of an equilibrium of the system or a level containing it) by means of billiards has also been answered in the affirmative. For singularities of centre-centre, centre-saddle, and saddle-saddle type the approach proposed and developed by Kibkalo [65], [66] uses confocal billiards (as limits of geodesic flows on an ellipsoid) and sets of permutations found by Oshemkov for saddle singularities of smooth systems. Focus-focus singularities were modelled by Vedyushkina, Kibkalo, and Pustovoitov [67] by means of billiard books with attracting Hooke potentials which are glued of n discs with permutation (1, .…”

Section: Integrable Generalizations Of Planar Billiards and Billiard ...mentioning

confidence: 99%

“…A simplest example id the Hooke potential ±k(x 2 + y 2 ) for a confocal or circular billiard table. One can also introduce such a potential on a billiard book glued of confocal (or circular) billiard domains [66]: the law of reflection from the boundary preserves the values of the additional integral and energy, the boundaries of domains of possible motion (the projections of Liouville leaves onto the table) lie on confocal quadrics (concentric circles), and the projection of the singular circle of each 3-atom for such a system also lies on some quadric. The situation is similar for an entire class of polynomial potentials (Pustovoitov).…”

Section: Integrable Billiards With Potentialmentioning

confidence: 99%

“…, n) models a focal singularity with n points of focus-focus type (Vedyushkina,Kibkalo,and Pustovoitov [67]). An arbitrary non-degenerate semilocal singularity of saddle-saddle or centre-saddle type is modelled by a book constructed algorithmically from this singularity and glued of sheets of type A ′ 0 (Kibkalo; see [65] and [66]). In [65] a limit transition (as the semiminor axis tends to zero) from the system of the flow on an ellipsoid E 2 ⊂ R 3 in a central repulsive or attractive Hooke potential to the topological billiard on a table 2A 2 with potential k(x 2 +y 2 )/2 was considered: the table is glued of two planar tables bounded by the same ellipse.…”

Section: 3mentioning

confidence: 99%

“…The problem of modelling non-degenerate semilocal singularities of rank 0, mentioned above, that is, of modelling Liouville foliations in a small four-dimensional neighbourhood of a level with non-degenerate equilibria, was successfully solved using an approach proposed by Kibkalo [65], [66]. To do this one uses billiard books on sheets of which motion proceed in the field of a repulsive Hooke potantial, the same for all sheets.…”

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: )mentioning

confidence: 99%

Section: Integrable Generalizations Of Planar Billiards and Billiard ...mentioning

confidence: 99%

Section: Integrable Billiards With Potentialmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Billiards and integrable systems

Fomenko,

Vedyushkina

2023

Russian Math. Surveys

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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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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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Saddle Singularities in Integrable Hamiltonian Systems: Examples and Algorithms

Cited by 12 publications

References 34 publications

Billiards and integrable systems

Billiards and integrable systems

Topology of Liouville foliations of integrable billiards on table-complexes

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

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