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TMF, 2002, Volume 131, Number 1, Pages 118–125 (Mi tmf318)  

This article is cited in 30 scientific papers (total in 30 papers)

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

V. V. Sokolova, A. V. Tsiganovb

a Landau Institute for Theoretical Physics, Centre for Non-linear Studies
b St. Petersburg State University, Faculty of Physics

Abstract: We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A $5\times 5$ matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the $so(p,q)$ Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov–Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its $3\times 3$ matrix Lax representation is also constructed.

DOI: https://doi.org/10.4213/tmf318

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English version:
Theoretical and Mathematical Physics, 2002, 131:1, 543–549

Bibliographic databases:

Received: 19.11.2001

Citation: V. V. Sokolov, A. V. Tsiganov, “Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops”, TMF, 131:1 (2002), 118–125; Theoret. and Math. Phys., 131:1 (2002), 543–549

Citation in format AMSBIB
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\jour Theoret. and Math. Phys.
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\pages 543--549
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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. Sokolov, A. V. Tsiganov, “Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models”, Theoret. and Math. Phys., 133:3 (2002), 1730–1743  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. 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
    3. V. V. Belov, S. Yu. Dobrokhotov, V. A. Maksimov, “Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application”, Theoret. and Math. Phys., 135:3 (2003), 765–791  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Wolf, T, “Classification of integrable quadratic Hamiltonians on e(3)”, Regular & Chaotic Dynamics, 8:2 (2003), 155  crossref  mathscinet  adsnasa  isi  scopus  scopus
    5. Komarov, IV, “Poisson maps and integrable deformations of the Kowalevski top”, Journal of Physics A-Mathematical and General, 36:29 (2003), 8035  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. I. Z. Golubchik, V. V. Sokolov, “Factorization of the Loop Algebra and Integrable Toplike Systems”, Theoret. and Math. Phys., 141:1 (2004), 1329–1347  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. A. V. Tsiganov, “Integrable Deformations of Tops Related to the Algebra $so(p,q)$”, Theoret. and Math. Phys., 141:1 (2004), 1348–1360  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. Tsiganov AV, Goremykin OV, “Integrable systems on so(4) related to XXX spin chains with boundaries”, Journal of Physics A-Mathematical and General, 37:17 (2004), 4843–4849  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. Sokolov VV, “One class of quadratic so(4) Hamiltonians”, Doklady Mathematics, 69:1 (2004), 108–111  zmath  isi
    10. Wolf T., “Applications of CRACK in the classification of integrable systems”, Superintegrability in Classical and Quantum Systems, CRM Proceedings & Lecture Notes, 37, 2004, 283–300  crossref  mathscinet  zmath  isi
    11. Tsiganov, AV, “A new integrable system on S-2 with the second integral quartic in the momenta”, Journal of Physics A-Mathematical and General, 38:16 (2005), 3547  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. Tsiganov, AV, “On a family of integrable systems on S-2 with a cubic integral of motion”, Journal of Physics A-Mathematical and General, 38:4 (2005), 921  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    13. Sokolov, VV, “Integrable quadratic classical Hamiltonians on so(4) and so(3,1)”, Journal of Physics A-Mathematical and General, 39:8 (2006), 1915  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    14. P. E. Ryabov, “Phase topology of one irreducible integrable problem in the dynamics of a rigid body”, Theoret. and Math. Phys., 176:2 (2013), 1000–1015  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    15. Yehia H.M., Elmandouh A.A., “A New Integrable Problem with a Quartic Integral in the Dynamics of a Rigid Body”, J. Phys. A-Math. Theor., 46:14 (2013), 142001  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    16. A. V. Vershilov, Yu. A. Grigorev, A. V. Tsyganov, “Ob odnoi integriruemoi deformatsii volchka Kovalevskoi”, Nelineinaya dinam., 10:2 (2014), 223–236  mathnet
    17. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Superintegrable Generalizations of the Kepler and Hook Problems”, Regul. Chaotic Dyn., 19:3 (2014), 415–434  mathnet  crossref  mathscinet  zmath
    18. 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
    19. A. V. Tsyganov, “Razdelenie peremennykh dlya odnogo obobscheniya sistemy Chaplygina na sfere”, Nelineinaya dinam., 11:1 (2015), 179–185  mathnet  elib
    20. Ryabov P.E., “New Invariant Relations For the Generalized Two-Field Gyrostat”, J. Geom. Phys., 87 (2015), 415–421  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    21. A. P. Sozonov, A. V. Tsiganov, “Bäcklund transformations relating different Hamilton–Jacobi equations”, Theoret. and Math. Phys., 183:3 (2015), 768–781  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    22. P. E. Ryabov, A. Yu. Savushkin, “Fazovaya topologiya volchka Kovalevskoi – Sokolova”, Nelineinaya dinam., 11:2 (2015), 287–317  mathnet
    23. 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
    24. Ol'shanskii V.Yu., “Linear invariant relations of Kirchhoff's equations”, Pmm-J. Appl. Math. Mech., 79:4 (2015), 334–349  crossref  mathscinet  isi  scopus
    25. 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
    26. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Generalizations of the Kovalevskaya case and quaternions”, Proc. Steklov Inst. Math., 295 (2016), 33–44  mathnet  crossref  crossref  mathscinet  isi  elib
    27. Yehia H.M., Elmandouh A.A., “A new conditional integrable case in the dynamics of a rigid body-gyrostat”, Mech. Res. Commun., 78:A (2016), 25–27  crossref  mathscinet  isi  scopus
    28. Oshemkov A.A. Ryabov P.E. Sokolov S.V., “Explicit Determination of Certain Periodic Motions of a Generalized Two-Field Gyrostat”, Russ. J. Math. Phys., 24:4 (2017), 517–525  crossref  mathscinet  zmath  isi  scopus  scopus
    29. Sokolov S.V., “New Invariant Relations For One Critical Subsystem of a Generalized Two-Field Gyrostat”, Dokl. Phys., 62:12 (2017), 567–570  crossref  isi  scopus  scopus
    30. Borisov A. Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520  mathscinet  isi
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