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 Mat. Sb., 2011, Volume 202, Number 5, Pages 127–160 (Mi msb7823)

Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$

D. V. Novikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The Sokolov integrable case on $\mathrm{e}(3)^{\star}$ is investigated. This is a Hamiltonian system with $2$ degrees of freedom in which the Hamiltonian and the additional integral are homogeneous polynomials having degree $2$ and $4$, respectively. This system is of interest because connected joint level surfaces of the Hamiltonian and the additional integral are noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed and the Liouville foliation of the system is described. The Hamiltonian vector fields corresponding to the Hamiltonian and the additional integral are proved to be complete.
Bibliography: 22 titles.

Keywords: integrable Hamiltonian systems, completeness of vector fields, bifurcation diagram, moment map, noncompact singularities.

DOI: https://doi.org/10.4213/sm7823

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English version:
Sbornik: Mathematics, 2011, 202:5, 749–781

Bibliographic databases:

UDC: 517.938.5
MSC: Primary 37J35; Secondary 70E40

Citation: D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$”, Mat. Sb., 202:5 (2011), 127–160; Sb. Math., 202:5 (2011), 749–781

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7823
• https://doi.org/10.4213/sm7823
• http://mi.mathnet.ru/eng/msb/v202/i5/p127

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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. Novikov D.V., “The topology of isoenergy surfaces for the Sokolov integrable case on the Lie algebra so(3,1)”, Moscow Univ. Math. Bull., 66:4 (2011), 181–184
2. D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{so}(3,1)$”, Sb. Math., 205:8 (2014), 1107–1132
3. D. A. Fedoseev, A. T. Fomenko, “Nekompaktnye osobennosti integriruemykh dinamicheskikh sistem”, Fundament. i prikl. matem., 21:6 (2016), 217–243
4. R. Akbarzadeh, “The topology of isoenergetic surfaces for the Borisov–Mamaev–Sokolov integrable case on the Lie algebra $so(3,1)$”, Theoret. and Math. Phys., 197:3 (2018), 1727–1736
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