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.
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).
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\by S.~S.~Nikolaenko
\paper Topological classification of the Goryachev integrable case in rigid body dynamics
\jour Sb. Math.
\yr 2016
\vol 207
\issue 1
\pages 113--139
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This publication is cited in the following 5 articles:
I. F. Kobtsev, “An elliptic billiard in a potential force field: classification of motions, topological analysis”, Sb. Math., 211:7 (2020), 987–1013
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
I. F. Kobtsev, “The geodesic flow on a two-dimensional ellipsoid in the field of an elastic force. Topological classification of solutions”, Moscow University Mathematics Bulletin, 73:2 (2018), 64–70
V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Math., 81:4 (2017), 688–733
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
Citation in format AMSBIB
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