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., 2000, Volume 191, Number 2, Pages 3–42 (Mi msb451)  

This article is cited in 43 scientific papers (total in 44 papers)

The method of loop molecules and the topology of the Kovalevskaya top

A. V. Bolsinova, P. H. Richterb, A. T. Fomenko

a M. V. Lomonosov Moscow State University
b University of Bremen, Institute for Theoretical Physics

Abstract: A method for calculating topological invariants of the foliation of a phase space into invariant Liouville tori in the case of integrable Hamiltonian systems with two degrees of freedom is put forward. The structure of this foliation is completely described for the Kovalevskaya integrable case in rigid body dynamics.


Full text: PDF file (758 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2000, 191:2, 151–188

Bibliographic databases:

UDC: 513.944+515.1
MSC: 58F07, 70E15
Received: 28.10.1999

Citation: A. V. Bolsinov, P. H. Richter, A. T. Fomenko, “The method of loop molecules and the topology of the Kovalevskaya top”, Mat. Sb., 191:2 (2000), 3–42; Sb. Math., 191:2 (2000), 151–188

Citation in format AMSBIB
\by A.~V.~Bolsinov, P.~H.~Richter, A.~T.~Fomenko
\paper The method of loop molecules and the~topology of the~Kovalevskaya top
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 2
\pages 3--42
\jour Sb. Math.
\yr 2000
\vol 191
\issue 2
\pages 151--188

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. V. V. Kozlov, “Sofya Kovalevskaya: a mathematician and a person”, Russian Math. Surveys, 55:6 (2000), 1175–1192  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Orel O.E., Ryabov P.E., “Topology, bifurcations and Liouville classification of Kirchhoff equations with an additional integral of fourth degree”, Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000), J. Phys. A, 34, no. 11, 2001, 2149–2163  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    3. P. V. Morozov, “The Liouville classification of integrable systems of the Clebsch case”, Sb. Math., 193:10 (2002), 1507–1533  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. T. G. Vozmischeva, A. A. Oshemkov, “Topological analysis of the two-centre problem on the two-dimensional sphere”, Sb. Math., 193:8 (2002), 1103–1138  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. P. E. Ryabov, “Bifurcations of First Integrals in the Sokolov Case”, Theoret. and Math. Phys., 134:2 (2003), 181–197  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. 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
    7. Gashenenko I.N., Richter P.H., “Enveloping surfaces and admissible velocities of heavy rigid bodies”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14:8 (2004), 2525–2553  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. 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
    9. Kharlamov M.P., “Bifurcation diagrams of the Kowalevski top in two constant fields”, Regul. Chaotic Dyn., 10:4 (2005), 381–398  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    10. D. B. Zot'ev, “Phase topology of Appelrot class I of Kowalewski top in a magnetic field”, J. Math. Sci., 149:1 (2008), 922–946  mathnet  crossref  mathscinet  zmath  elib
    11. P. V. Morozov, “Calculation of the Fomenko–Zieschang invariants in the Kovalevskaya–Yehia integrable case”, Sb. Math., 198:8 (2007), 1119–1143  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. Radnović M., Rom-Kedar V., “Foliations of isonergy surfaces and singularities of curves”, Regul. Chaotic Dyn., 13:6 (2008), 645–668  mathnet  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    13. E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko, “Maximally symmetric cell decompositions of surfaces and their coverings”, Sb. Math., 199:9 (2008), 1263–1353  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. M. P. Kharlamov, “Topologicheskii analiz i bulevy funktsii: I. Metody i prilozheniya k klassicheskim sistemam”, Nelineinaya dinam., 6:4 (2010), 769–805  mathnet
    15. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and stability of integrable systems”, Russian Math. Surveys, 65:2 (2010), 259–318  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. M. P. Kharlamov, P. E. Ryabov, “Diagrammy Smeila–Fomenko i grubye invarianty sluchaya Kovalevskoi–Yakhya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2011, no. 4, 40–59  mathnet
    17. Pelayo A., San Vu Ngoc, “Symplectic Theory of Completely Integrable Hamiltonian Systems”, Bull Amer Math Soc, 48:3 (2011), 409–455  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    18. Novikov D.V., “Topologiya izoenergeticheskikh poverkhnostei dlya integriruemogo sluchaya sokolova na algebre li so(3,1)}”, Vestnik Moskovskogo universiteta. Seriya 1: Matematika. Mekhanika, 2011, no. 4, 62–65  mathnet  elib
    19. D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$”, Sb. Math., 202:5 (2011), 749–781  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. D. B. Zot'ev, “Fomenko–Zieschang invariants of integrable systems with symplectic singularities”, Russian Math. (Iz. VUZ), 56:1 (2012), 19–26  mathnet  crossref  mathscinet
    21. N. S. Logacheva, “Classification of nondegenerate equilibria and degenerate 1-dimensional orbits of the Kovalevskaya-Yehia integrable system”, Sb. Math., 203:1 (2012), 28–59  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    22. P. P. Andreyanov, K. E. Dushin, “Bifurcation sets in the Kovalevskaya-Yehia problem”, Sb. Math., 203:4 (2012), 459–499  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. P. E. Ryabov, M. P. Kharlamov, “Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field”, Sb. Math., 203:2 (2012), 257–287  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    24. Slavina N.S., “Classification of the Family of Kovalevskaya-Yehia Systems Up to Liouville Equivalence”, Dokl. Math., 88:2 (2013), 537–540  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    25. M.P. Kharlamov, “Phase topology of one system with separated variables and singularities of the symplectic structure”, Journal of Geometry and Physics, 2014  crossref  mathscinet  scopus  scopus  scopus
    26. Mikhail P. Kharlamov, “Extensions of the Appelrot Classes for the Generalized Gyrostat in a Double Force Field”, Regul. Chaotic Dyn., 19:2 (2014), 226–244  mathnet  crossref  mathscinet  zmath
    27. I. K. Kozlov, “The topology of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)”, Sb. Math., 205:4 (2014), 532–572  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. S. S. Nikolaenko, “A topological classification of the Chaplygin systems in the dynamics of a rigid body in a fluid”, Sb. Math., 205:2 (2014), 224–268  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    29. D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{so}(3,1)$”, Sb. Math., 205:8 (2014), 1107–1132  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    30. N. S. Slavina, “Topological classification of systems of Kovalevskaya-Yehia type”, Sb. Math., 205:1 (2014), 101–155  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    31. M. P. Kharlamov, P. E. Ryabov, “Topological atlas of the Kovalevskaya top in a double field”, J. Math. Sci., 223:6 (2017), 775–809  mathnet  crossref  mathscinet  elib
    32. Rasoul Akbarzadeh, Ghorbanali Haghighatdoost, “The Topology of Liouville Foliation for the BorisovMamaevSokolov Integrable Case on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 20:3 (2015), 317–344  mathnet  crossref  mathscinet  zmath  adsnasa
    33. E. O. Kantonistova, “Liouville classification of integrable Hamiltonian systems on surfaces of revolution”, Moscow University Mathematics Bulletin, 70:5 (2015), 220–222  mathnet  crossref  mathscinet  isi
    34. 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
    35. Rasoul Akbarzadeh, “Topological Analysis Corresponding to the BorisovMamaevSokolov Integrable System on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 21:1 (2016), 1–17  mathnet  crossref  mathscinet  zmath
    36. Mikhail P. Kharlamov, Pavel E. Ryabov, Alexander Yu. Savushkin, “Topological Atlas of the KowalevskiSokolov Top”, Regul. Chaotic Dyn., 21:1 (2016), 24–65  mathnet  crossref  mathscinet  zmath
    37. 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  mathnet  crossref  mathscinet  isi
    38. El-Sabaa F.M. Hosny M. Zakria S.K., “Bifurcations of Liouville Tori of a Two Fixed Center Problem”, Astrophys. Space Sci., 363:4 (2018), 77  crossref  mathscinet  isi  scopus  scopus  scopus
    39. Bolsinov A., Guglielmi L., Kudryavtseva E., “Symplectic Invariants For Parabolic Orbits and Cusp Singularities of Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170424  crossref  mathscinet  zmath  isi  scopus
    40. V. V. Vedyushkina, I. S. Kharcheva, “Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems”, Sb. Math., 209:12 (2018), 1690–1727  mathnet  crossref  crossref  adsnasa  isi  elib
    41. Kibkalo V., “Topological Analysis of the Liouville Foliation For the Kovalevskaya Integrable Case on the Lie Algebra So(4)”, Lobachevskii J. Math., 39:9 (2018), 1396–1399  crossref  mathscinet  zmath  isi  scopus
    42. 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  mathnet  crossref  crossref  adsnasa  isi  elib
    43. A. T. Fomenko, V. V. Vedyushkina, “Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: recent results”, Theor. Appl. Mech., 46:1 (2019), 47–63  mathnet  crossref
    44. A. I. Zhila, “Klassifikatsiya matrits skleiki na krugovykh molekulakh tochek tipa tsentr-tsentr”, Fundament. i prikl. matem., 22:6 (2019), 85–94  mathnet
  •   - 1992–2005 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:911
    Full text:297
    First page:5

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2021