Phase topology of one irreducible integrable problem in the dynamics of a rigid body

Ryabov, Pavel E.

doi:10.1007/s11232-013-0087-0

Cited by 10 publications

(19 citation statements)

References 15 publications

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“…In the following sections we point out the equations of three invariant manifolds generalizing the Appelrot classes for the Hamilton function (2.13). Two of them exist only for the top with λ = 0 and correspond to the bifurcation surfaces found in [28], the third one exists for arbitrary values of λ and generalizes the cases shown in the works [18,19,17].…”

Section: The Appelrot Classesmentioning

confidence: 55%

“…P.E. Ryabov in [28] started the topological analysis of the GTFG system. First of all, he obtains the explicit formulas of the commutating integrals K and G generalizing K Y and G RS , thus expressing in terms of these integrals the coefficients of the algebraic curve associated with the Lax representation:…”

Section: Preliminariesmentioning

confidence: 99%

“…Next, in the work [28] for the case λ = 0 four critical subsystems of rank 2 are pointed out. For these subsystems, the constraints on the integral constants are derived from the condition that the algebraic curve of the Lax pair has a singular point.…”

Section: Preliminariesmentioning

confidence: 99%

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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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Section: The Appelrot Classesmentioning

confidence: 55%

Section: Preliminariesmentioning

confidence: 99%

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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 Hamiltonian function (2), we represent the additional integrals K and G as functions of two deformation parameters ε 1 and ε 2 [4]:…”

Section: Intrtoductionmentioning

confidence: 99%

Explicit determination of certain periodic motions of a generalized two-field gyrostat

Oshemkov

Ryabov

Sokolov

2017

Russ. J. Math. Phys.

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The case of motion of a generalized two-field gyrostat found by V. V. Sokolov and A. V. Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. We find a set of points at which the momentum map has rank 1. This set consists of special periodic motions which correspond to the singular points of a bifurcation diagram on an iso-energetic surface. For such motions the phase variables can be expressed in terms of algebraic functions of a single auxiliary variable. These algebraic functions satisfy a differential equation integrable in elliptic functions of time. It is shown that the corresponding points in the three-dimensional space of the constants of the integrals belong to the intersection of two sheets of the discriminant surface of the Lax curve.

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“…Remark 4. We note that, generally speaking, the authors of [19] presented a more general case of the Kovalevskaya top in which the equations of motion possess an additional integral of degree 2 and 4 in momenta (the explicit form of the integral is presented in [18]).…”

Section: The Kovalevskaya Casementioning

confidence: 99%

Generalizations of the Kovalevskaya case and quaternions

Bizyaev

Borisov

Mamaev

2016

Proc. Steklov Inst. Math.

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Abstract. This paper provides a detailed description of various reduction schemes in rigid body dynamics. Analysis of one of such nontrivial reductions makes it possible to order the cases already found and to obtain new generalizations of the Kovalevskaya case to e(3). We note that the above reduction allows one to obtain in a natural way some singular additive terms which were proposed earlier by D. N. Goryachev.

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Phase topology of one irreducible integrable problem in the dynamics of a rigid body

Cited by 10 publications

References 15 publications

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

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

Explicit determination of certain periodic motions of a generalized two-field gyrostat

Generalizations of the Kovalevskaya case and quaternions

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