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 Mat. Sb., 1995, Volume 186, Number 2, Pages 105–128 (Mi msb16)

Rotation function for integrable problems reducing to the Abel equations. Orbital classification of Goryachev–Chaplygin systems.

O. E. Orel

M. V. Lomonosov Moscow State University

Abstract: The rotation vector, one of the most important orbital invariants of integrable Hamiltonian systems with two degrees of freedom, is constructed using the rotation function (see [1]). A general theory of computation of the rotation functions for dynamical systems reducing to the Abel equations is developed. Using this theory, an explicit formula for the rotation function in the Goryachev–Chaplygin case in the dynamics of heavy rigid bodies is obtained. The orbital classification of the family of Goryachev–Chaplygin systems for different values of energy is presented. For this purpose the orbital Fomenko–Bolsinov invariant, which is the classifying object, is computed. The orbital non-equivalence of the Goryachev–Chaplygin flows on surfaces of equal energy corresponding to different energy levels is established as a result of an analytic study and computer analysis (together with S. Takahasi, Japan) . In addition, explicit formulae for the transition from the coordinate system on the Jacobian (the Abelian variables) to the Euler–Poisson coordinate system in the Goryachev–Chaplygin case are obtained and the covering of the Jacobian by the Liouville torus is studied. This can be used for finding an explicit solution to the Goryachev–Chaplygin problem in terms of two-dimensional theta functions.

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English version:
Sbornik: Mathematics, 1995, 186:2, 271–296

Bibliographic databases:

UDC: 514.745.82
MSC: 70H05, 58F05

Citation: O. E. Orel, “Rotation function for integrable problems reducing to the Abel equations. Orbital classification of Goryachev–Chaplygin systems.”, Mat. Sb., 186:2 (1995), 105–128; Sb. Math., 186:2 (1995), 271–296

Citation in format AMSBIB
\Bibitem{Ore95} \by O.~E.~Orel \paper Rotation function for integrable problems reducing to the~Abel equations. Orbital classification of Goryachev--Chaplygin systems. \jour Mat. Sb. \yr 1995 \vol 186 \issue 2 \pages 105--128 \mathnet{http://mi.mathnet.ru/msb16} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1330593} \zmath{https://zbmath.org/?q=an:0858.70010} \transl \jour Sb. Math. \yr 1995 \vol 186 \issue 2 \pages 271--296 \crossref{https://doi.org/10.1070/SM1995v186n02ABEH000016} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RZ91900016} 

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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. O. E. Orel, S. Takahashi, “Orbital classification of the integrable problems of Lagrange and Goryachev–Chaplygin by the methods of computer analysis”, Sb. Math., 187:1 (1996), 93–110
2. A. V. Bolsinov, “Fomenko invariants in the theory of integrable Hamiltonian systems”, Russian Math. Surveys, 52:5 (1997), 997–1015
3. P. V. Morozov, “Topology of Liouville foliations in the Steklov and the Sokolov integrable cases of Kirchhoff's equations”, Sb. Math., 195:3 (2004), 369–412
4. Dullin, HR, “A new integrable system on the sphere”, Mathematical Research Letters, 11:5–6 (2004), 715
5. Fomenko A.T., Morozov P.V., “Some new results in topological classification of integrable systems in rigid body dynamics”, Proceedings of the Workshop on Contemporary Geometry and Related Topics, 2004, 201–222
6. M. P. Kharlamov, “Topologicheskii analiz i bulevy funktsii: I. Metody i prilozheniya k klassicheskim sistemam”, Nelineinaya dinam., 6:4 (2010), 769–805
7. Fomenko A.T. Nikolaenko S.S., “The Chaplygin Case in Dynamics of a Rigid Body in Fluid Is Orbitally Equivalent To the Euler Case in Rigid Body Dynamics and To the Jacobi Problem About Geodesics on the Ellipsoid”, J. Geom. Phys., 87 (2015), 115–133
8. S. S. Nikolaenko, “Topological classification of the Goryachev integrable case in rigid body dynamics”, Sb. Math., 207:1 (2016), 113–139
9. V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Math., 81:4 (2017), 688–733
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