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 Mat. Sb., 2014, Volume 205, Number 1, Pages 105–160 (Mi msb8241)

Topological classification of systems of Kovalevskaya-Yehia type

N. S. Slavina

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

Abstract: All the Fomenko-Zieschang invariants are calculated for the Kovalevskaya-Yehia problem, for all noncritical values of the parameters $g$ and $\lambda$, by constructing admissible systems of coordinates and determining the mutual disposition of the basis cycles. The family of Kovalevskaya-Yehia systems contains 29 pairwise Liouville non-equivalent foliations. These foliations include those that are equivalent to previously known foliations, which arose in the integrable cases of Kovalevskaya and of Kovalevskaya-Yehia for $g=0$, in the Zhukovskiǐ case, and in the Goryachev-Chaplygin-Sretenskiǐ case. Eleven new foliations are included in the 29 foliations, new in the sense that they are not Liouville equivalent to any foliations discovered earlier which arose in the known integrable cases of the rigid body. The topological type of the Liouville foliation for the family of Kovalevskaya-Yehia systems stabilizes at large values of the energy $H$, and this ‘high-energy’ system is roughly Liouville equivalent, at one of the energy levels, to the Goryachev-Chaplygin-Sretenskiǐ integrable case, which is already known.
Bibliography: 29 titles.

Keywords: Fomenko-Zieschang invariants, Liouville foliation, Liouville equivalent integrable systems.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-00664a Ministry of Education and Science of the Russian Federation ÍØ-1410.2012.1

DOI: https://doi.org/10.4213/sm8241

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English version:
Sbornik: Mathematics, 2014, 205:1, 101–155

Bibliographic databases:

UDC: 517.938.5
MSC: Primary 37J35; Secondary 70E17

Citation: N. S. Slavina, “Topological classification of systems of Kovalevskaya-Yehia type”, Mat. Sb., 205:1 (2014), 105–160; Sb. Math., 205:1 (2014), 101–155

Citation in format AMSBIB
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• https://doi.org/10.4213/sm8241
• http://mi.mathnet.ru/eng/msb/v205/i1/p105

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. A. Kibkalo, “The topology of the analog of Kovalevskaya integrability case on the Lie algebra $\mathrm{so}(4)$ under zero area integral”, Moscow University Mathematics Bulletin, 71:3 (2016), 119–123
2. V. V. Fokicheva, A. T. Fomenko, “Billiard systems as the models for the rigid body dynamics”, Advances in dynamical systems and control, Stud. Syst. Decis. Control, 69, ed. V. Sadovnichiy, M. Zgurovsky, Springer, Cham, 2016, 13–33
3. V. A. Kibkalo, “Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra $\operatorname{so}(4)$”, Sb. Math., 210:5 (2019), 625–662
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