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Mat. Sb., 2016, Volume 207, Number 1, Pages 123–150 (Mi msb8520)  

This article is cited in 5 scientific papers (total in 5 papers)

Topological classification of the Goryachev integrable case in rigid body dynamics

S. S. Nikolaenko

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A topological analysis of the Goryachev integrable case in rigid body dynamics is made on the basis of the Fomenko-Zieschang theory. The invariants (marked molecules) which are obtained give a complete description, from the standpoint of Liouville classification, of the systems of Goryachev type on various level sets of the energy. It turns out that on appropriate energy levels the Goryachev case is Liouville equivalent to many classical integrable systems and, in particular, the Joukowski, Clebsch, Sokolov and Kovalevskaya-Yehia cases in rigid body dynamics, as well as to some integrable billiards in plane domains bounded by confocal quadrics — in other words, the foliations given by the closures of generic solutions of these systems have the same structure.
Bibliography: 15 titles.

Keywords: integrable Hamiltonian system, topological classification, Liouville foliation, Goryachev case, marked molecule.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00664а
Ministry of Education and Science of the Russian Federation НШ-581.2014.1
This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 13-01-00664a) and the Programme of the President of the Russian Federation for the Support of Leading Scientific Schools (grant no. НШ-581.2014.1).


DOI: https://doi.org/10.4213/sm8520

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English version:
Sbornik: Mathematics, 2016, 207:1, 113–139

Bibliographic databases:

UDC: 517.938.5
MSC: Primary 37J35, 70E40; Secondary 37N10
Received: 25.03.2015 and 18.06.2015

Citation: S. S. Nikolaenko, “Topological classification of the Goryachev integrable case in rigid body dynamics”, Mat. Sb., 207:1 (2016), 123–150; Sb. Math., 207:1 (2016), 113–139

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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. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Math., 81:4 (2017), 688–733  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. S. S. Nikolaenko, “Topological classification of the Goryachev integrable systems in the rigid body dynamics: non-compact case”, Lobachevskii J. Math., 38:6 (2017), 1050–1060  crossref  mathscinet  zmath  isi  scopus
    3. I. F. Kobtsev, “Geodesic flow of a 2D ellipsoid in an elastic stress field: topological classification of solutions”, Moscow University Mathematics Bulletin, 73:2 (2018), 64–70  mathnet  crossref  mathscinet  zmath  isi
    4. V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Math., 83:6 (2019), 1137–1173  mathnet  crossref  crossref  adsnasa  isi  elib
    5. I. F. Kobtsev, “An elliptic billiard in a potential force field: classification of motions, topological analysis”, Sb. Math., 211:7 (2020), 987–1013  mathnet  crossref  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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