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TMF, 2013, Volume 176, Number 2, Pages 205–221 (Mi tmf8515)  

This article is cited in 8 scientific papers (total in 8 papers)

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

P. E. Ryabov

Financial University, Moscow, Russia

Abstract: We consider the integrable system with three degrees of freedom for which V. V. Sokolov and A. V. Tsiganov specified the Lax pair. The Lax representation generalizes the $L$$A$ pair found by A. G. Reyman and M. A. Semenov-Tian-Shansky for the Kovalevskaya gyrostat in a double field. We give explicit formulas for the additional first integrals $K$ and $G$ (independent almost everywhere), which are functionally related to the coefficients of the spectral curve for the Sokolov–Tsiganov $L$$A$ pair. Using this form of the additional integrals $K$ and $G$ and the Kharlamov parametric reduction, we analytically present two invariant four-dimensional submanifolds where the induced dynamical system is Hamiltonian (almost everywhere) with two degrees of freedom. The system of equations specifying one of the invariant submanifolds is a generalization of the invariant relations for the integrable Bogoyavlensky case (rotation of a magnetized rigid body in homogeneous gravitational and magnetic fields). We use the method of critical subsystems to describe the phase topology of the whole system. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of the Liouville tori both inside the subsystems and in the whole system.

Keywords: completely integrable Hamiltonian system, spectral curve, moment map, bifurcation diagram, bifurcation of Liouville tori

DOI: https://doi.org/10.4213/tmf8515

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English version:
Theoretical and Mathematical Physics, 2013, 176:2, 1000–1015

Bibliographic databases:

Received: 13.02.2013

Citation: P. E. Ryabov, “Phase topology of one irreducible integrable problem in the dynamics of a rigid body”, TMF, 176:2 (2013), 205–221; Theoret. and Math. Phys., 176:2 (2013), 1000–1015

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Vershilov, Yu. A. Grigorev, A. V. Tsyganov, “Ob odnoi integriruemoi deformatsii volchka Kovalevskoi”, Nelineinaya dinam., 10:2 (2014), 223–236  mathnet
    2. Mikhail P. Kharlamov, “Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field”, Regul. Chaotic Dyn., 19:2 (2014), 226–244  mathnet  crossref  mathscinet  zmath
    3. P. E. Ryabov, “New invariant relations for the generalized two-field gyrostat”, J. Geom. Phys., 87 (2015), 415–421  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. P. E. Ryabov, A. Yu. Savushkin, “Fazovaya topologiya volchka Kovalevskoi – Sokolova”, Nelineinaya dinam., 11:2 (2015), 287–317  mathnet
    5. Mikhail P. Kharlamov, Pavel E. Ryabov, Alexander Yu. Savushkin, “Topological Atlas of the KowalevskiSokolov Top”, Regul. Chaotic Dyn., 21:1 (2016), 24–65  mathnet  crossref  mathscinet  zmath
    6. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Generalizations of the Kovalevskaya case and quaternions”, Proc. Steklov Inst. Math., 295 (2016), 33–44  mathnet  crossref  crossref  mathscinet  isi  elib
    7. A. A. Oshemkov, P. E. Ryabov, S. V. Sokolov, “Explicit determination of certain periodic motions of a generalized two-field gyrostat”, Russ. J. Math. Phys., 24:4 (2017), 517–525  crossref  mathscinet  zmath  isi  scopus
    8. S. V. Sokolov, “New invariant relations for one critical subsystem of a generalized two-field gyrostat”, Dokl. Phys., 62:12 (2017), 567–570  crossref  isi  scopus
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