Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field

Ryabov, Pavel E.; Kharlamov, Mikhail P.

doi:10.1070/sm2012v203n02abeh004222

Cited by 7 publications

(12 citation statements)

References 24 publications

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“…The description of the diagrams Σ * i and the bifurcations inside the critical subsystems is completed in the works [27,28,32,29]. The classification of the points of the set K with respect to the complete initial system with three degrees of freedom on P 6 is obtained in the work [30]. In the next two sections we give a short exposition of the necessary results of the above cited papers.…”

Section: Equations and Integrals The Notion Of A Critical Subsystemmentioning

confidence: 99%

“…As was mentioned above, all the points of the trajectories in the sets of degeneration of the forms induced by the symplectic structure are degenerate critical points of the momentum map F for any rank. The formal proof is given in [30]. There are no such points among the points of rank 0.…”

Section: (34)mentioning

confidence: 99%

“…For the proof, in [30] the explicit form of the characteristic polynomials of the symplectic operator A H are calculated at the points p i .…”

Section: Classification Of Critical Points By Typementioning

confidence: 99%

“…For the proof of the theorem, in the work [30], the pairs of the first integrals generating the Cartan subalgebra of dimension 2 in the algebra of the symplectic operators are explicitly written out and it is shown that the following integrals generate regular elements in these subalgebras:…”

Section: Classification Of Critical Points By Typementioning

confidence: 99%

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Topological Atlas of the Kovalevskaya Top in a Double Field

Kharlamov

Ryabov

2017

J Math Sci

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We fulfill the rough topological analysis of the problem of the motion of the Kovalevskaya top in a double field. This problem is described by a completely integrable system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The notion of a topological atlas of an irreducible system is introduced. The complete topological analysis of the critical subsystems with two degrees of freedom is given. We calculate the types of all critical points. We present the parametric classification of the equipped iso-energy diagrams of the complete momentum map pointing out all chambers, families of 3-tori, and 4-atoms of their bifurcations. Basing on the ideas of A.T. Fomenko, we introduce the notion of the simplified net iso-energy invariant. All such invariants are constructed. Using them, we establish, for all parametrically stable cases, the number of critical periodic solutions of all types and the loop molecules of all rank 1 singularities.

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Section: Equations and Integrals The Notion Of A Critical Subsystemmentioning

confidence: 99%

Section: (34)mentioning

confidence: 99%

“…For the proof, in [30] the explicit form of the characteristic polynomials of the symplectic operator A H are calculated at the points p i .…”

Section: Classification Of Critical Points By Typementioning

confidence: 99%

Section: Classification Of Critical Points By Typementioning

confidence: 99%

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Topological Atlas of the Kovalevskaya Top in a Double Field

Kharlamov

Ryabov

2017

J Math Sci

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“…The knowledge of critical subsystems made it possible to obtain many analytical results (e.g, new separation of variables [22,18], explicit descriptions of periodic motions [19,20]) and to fulfill a number of topological investigations [22,18,23]. On this basis, the complete description of the topology of the Kowalevski top in a double field in terms of net topological invariants was given [24,25].…”

Section: Introductionmentioning

confidence: 99%

Extensions of the Appelrot classes for the generalized gyrostat in a double force field

Kharlamov

2014

Regul. Chaot. Dyn.

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For the integrable system on e(3, 2) found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.MSC 70E05,70E17,37J15, 37J20

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Qualitative Analysis of the Reyman – Semenov–Tian–Shansky Integrable Case of the Generalized Kowalewski Top

Irtegov

Titorenko

2016

Computer Algebra in Scientific Computing

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Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field

Cited by 7 publications

References 24 publications

Topological Atlas of the Kovalevskaya Top in a Double Field

Topological Atlas of the Kovalevskaya Top in a Double Field

Extensions of the Appelrot classes for the generalized gyrostat in a double force field

Qualitative Analysis of the Reyman – Semenov–Tian–Shansky Integrable Case of the Generalized Kowalewski Top

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