General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mat. Sb.:

Personal entry:
Save password
Forgotten password?

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

This article is cited in 11 scientific papers (total in 11 papers)

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.

Full text: PDF file (2890 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 514.745.82
MSC: 70H05, 58F05
Received: 03.10.1994

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
\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
\jour Sb. Math.
\yr 1995
\vol 186
\issue 2
\pages 271--296

Linking options:

    SHARE: FaceBook Twitter Livejournal

    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. V. Bolsinov, “Fomenko invariants in the theory of integrable Hamiltonian systems”, Russian Math. Surveys, 52:5 (1997), 997–1015  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Dullin, HR, “A new integrable system on the sphere”, Mathematical Research Letters, 11:5–6 (2004), 715  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi
    6. M. P. Kharlamov, “Topologicheskii analiz i bulevy funktsii: I. Metody i prilozheniya k klassicheskim sistemam”, Nelineinaya dinam., 6:4 (2010), 769–805  mathnet
    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  crossref  mathscinet  zmath  isi
    8. S. S. Nikolaenko, “Topological classification of the Goryachev integrable case in rigid body dynamics”, Sb. Math., 207:1 (2016), 113–139  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. “Billiard Systems as the Models For the Rigid Body Dynamics”, Advances in Dynamical Systems and Control, Studies in Systems Decision and Control, 69, 2016, 13–33  crossref  mathscinet  zmath  isi
    10. 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
    11. Ivan Yu. Polekhin, “Precession of the Kovalevskaya and Goryachev – Chaplygin Tops”, Regul. Chaotic Dyn., 24:3 (2019), 281–297  mathnet  crossref  mathscinet
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:295
    Full text:101
    First page:2

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020