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 Mat. Sb., 2012, Volume 203, Number 2, Pages 111–142 (Mi msb7758)

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

P. E. Ryabova, M. P. Kharlamovb

a Financial University under the Government of the Russian Federation, Moscow

Abstract: The problem of motion of the Kovalevskaya top in a double force field is investigated (the integrable case of A. G. Reyman and M. A. Semenov-Tian-Shansky without a gyrostatic momentum). It is a completely integrable Hamiltonian system with three degrees of freedom not reducible to a family of systems with two degrees of freedom. The critical set of the integral map is studied. The critical subsystems and bifurcation diagrams are described. The classification of all nondegenerate critical points is given. The set of these points consists of equilibria (nondegenerate singularities of rank 0), of singular periodic motions (nondegenerate singularities of rank 1), and also of critical two-frequency motions (nondegenerate singularities of rank 2).
Bibliography: 32 titles.

Keywords: singularities of integrable Hamiltonian systems, momentum map, bifurcation diagram.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm7758

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English version:
Sbornik: Mathematics, 2012, 203:2, 257–287

Bibliographic databases:

UDC: 517.938.5+531.38
MSC: Primary 70E40; Secondary 37J35, 37N05, 70G40

Citation: P. E. Ryabov, M. P. Kharlamov, “Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field”, Mat. Sb., 203:2 (2012), 111–142; Sb. Math., 203:2 (2012), 257–287

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7758
• https://doi.org/10.4213/sm7758
• http://mi.mathnet.ru/eng/msb/v203/i2/p111

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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. Kharlamov M.P., Ryabov P.E., “Net diagrams for the Fomenko invariant in the integrable system with three degrees of freedom”, Dokl. Math., 86:3 (2012), 839–842
2. P. E. Ryabov, “Phase topology of one irreducible integrable problem in the dynamics of a rigid body”, Theoret. and Math. Phys., 176:2 (2013), 1000–1015
3. 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
4. M. P. Kharlamov, “Phase topology of one system with separated variables and singularities of the symplectic structure”, J. Geom. Phys., 87 (2015), 248–265
5. M. P. Kharlamov, P. E. Ryabov, “Topological atlas of the Kovalevskaya top in a double field”, J. Math. Sci., 223:6 (2017), 775–809
6. V. Irtegov, T. Titorenko, “Qualitative analysis of the Reyman–Semenov–Tian–Shansky integrable case of the generalized Kowalewski top”, Computer algebra in scientific computing, Lecture Notes in Comput. Sci., 9890, eds. V. Gerdt, W. Koepf, W. Seiler, E. Vorozhtsov, Springer, Cham, 2016, 289–304
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