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 Regul. Chaotic Dyn., 2014, Volume 19, Issue 2, Pages 226–244 (Mi rcd128)

Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field

Mikhail P. Kharlamov

Abstract: 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.

Keywords: generalized two-field gyrostat, critical subsystems, Appelrot classes, invariant relations, types of critical points

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-9702514-01-00119 This work was partially supported by RFBR and the authorities of the Volgograd Region, research projects No. 13-01-97025, 14-01-00119.

DOI: https://doi.org/10.1134/S1560354714020063

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MSC: 70E05, 70E17, 37J15, 37J20
Accepted:30.10.2013
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Citation: 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

Citation in format AMSBIB
\Bibitem{Kha14} \by Mikhail~P.~Kharlamov \paper Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field \jour Regul. Chaotic Dyn. \yr 2014 \vol 19 \issue 2 \pages 226--244 \mathnet{http://mi.mathnet.ru/rcd128} \crossref{https://doi.org/10.1134/S1560354714020063} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3189259} \zmath{https://zbmath.org/?q=an:1309.70007} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000334198000006} 

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• http://mi.mathnet.ru/eng/rcd/v19/i2/p226

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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. P. E. Ryabov, A. Yu. Savushkin, “Fazovaya topologiya volchka Kovalevskoi – Sokolova”, Nelineinaya dinam., 11:2 (2015), 287–317
2. P. E. Ryabov, “New invariant relations for the generalized two-field gyrostat”, J. Geom. Phys., 87 (2015), 415–421
3. Mikhail P. Kharlamov, Pavel E. Ryabov, Alexander Yu. Savushkin, “Topological Atlas of the Kowalevski–Sokolov Top”, Regul. Chaotic Dyn., 21:1 (2016), 24–65
4. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Generalizations of the Kovalevskaya case and quaternions”, Proc. Steklov Inst. Math., 295 (2016), 33–44
5. P. E. Ryabov, S. V. Sokolov, “Phase Topology of Two Vortices of Identical Intensities in a Bose – Einstein Condensate”, Nelineinaya dinam., 15:1 (2019), 59–66
6. P. E. Ryabov, “Bifurcations of Liouville tori in a system of two vortices of positive intensity in a Bose–Einstein condensate”, Dokl. Math., 99:2 (2019), 225–229