Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra so(3, 1)

Kibkalo, Vladislav

doi:10.1016/j.topol.2019.107028

Cited by 13 publications

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References 16 publications

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“…The same can be said about integrable geodesic flows on 2-surfaces of revolution with a potential or a magnetic field added [84]- [87]. A Liouville classification of multiparameter analogues of the Kovalevskaya system on Lie algebras was obtained [88]- [93]. Also, pseudo-Euclidean analogues of mechanical systems introduced in [94] and, in particular, an analogue of the Kovalevskaya system [95] are under investigation.…”

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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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confidence: 99%

“…Note that the case κ ̸ = 0 corresponds to analogues of mechanical systems on other Lie algebras: on so(3, 1) and on so(4). Such systems have also been closely investigated from the standpoint of Liouville foliations for them (see [88]- [91], [93], [114], and [115]).…”

Section: Table 1 Combinations Of Marks R and ε On Edges Of Marked Mol...mentioning

confidence: 99%

Billiards and integrable systems

Fomenko,

Vedyushkina

2023

Russian Math. Surveys

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“…The initial (starting) state is a nonconnected billiard having no common points with the focal line (bottom). It is homeomorphic to two disks and realizes the Euler system on a pair of 3-spheres 2S 3 . Then it transforms into an annulus realizing a product of a 2-sphere and a circle: S 1 × S 2 .…”

Section: Definition 7 a Point Of The Phase Complex Tx(d I ) Is A Pairmentioning

confidence: 99%

“…Then it transforms into an annulus realizing a product of a 2-sphere and a circle: S 1 × S 2 . Next, the annulus turns into a sphere (ellipsoid) and realizes a projective space RP 3 . Figure 1b also shows the trajectories of the billiard ball and the gluing of spines.…”

Section: Definition 7 a Point Of The Phase Complex Tx(d I ) Is A Pairmentioning

confidence: 99%

“…It is well known that many IHS in mechanics, physics, and their analogues on Lie algebras were previously characterized by a "code," namely, by Fomenko-Zieschang invariants [1] of Liouville foliations of a system with constraints imposed on the set of isoenergy 3surfaces Q 3 (arranged in increasing order of the energy H) which represent regular energy zones of the system (see [1][2][3]).…”

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