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TMF, 2002, Volume 133, Number 3, Pages 485–500 (Mi tmf413)  

This article is cited in 14 scientific papers (total in 14 papers)

Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models

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 construct hierarchies of commutative Poisson subalgebras for Sklyanin brackets. Each of the subalgebras is generated by a complete set of integrals in involution. Some new integrable systems and schemes for separation of variables for them are elaborated using various well-known representations of the brackets. The constructed models include deformations for the Goryachev–Chaplygin top, the Toda chain, and the Heisenberg model.

Keywords: finite-dimensional integrable systems, Lax representation, $r$-matrix algebras, separation of variables

DOI: https://doi.org/10.4213/tmf413

Full text: PDF file (274 kB)
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English version:
Theoretical and Mathematical Physics, 2002, 133:3, 1730–1743

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Citation: V. V. Sokolov, A. V. Tsiganov, “Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models”, TMF, 133:3 (2002), 485–500; Theoret. and Math. Phys., 133:3 (2002), 1730–1743

Citation in format AMSBIB
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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. A. V. Tsiganov, “Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat”, Theoret. and Math. Phys., 135:2 (2003), 651–658  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. V. Tsiganov, “Toda Chains in the Jacobi Method”, Theoret. and Math. Phys., 139:2 (2004), 636–653  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Borisov AV, Mamaev IS, “Necessary and sufficient conditions for the polynomial integrability of generalized Toda chains”, Doklady Mathematics, 69:1 (2004), 131–135  mathscinet  zmath  isi
    4. 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
    5. A. V. Tsiganov, “Darboux–Nijenhuis variables for open generalized Toda chains”, Theoret. and Math. Phys., 152:3 (2007), 1243–1257  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Tsiganov, AV, “On maximally superintegrable systems”, Regular & Chaotic Dynamics, 13:3 (2008), 178  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Tsiganov, AV, “The Poisson bracket compatible with the classical reflection equation algebra”, Regular & Chaotic Dynamics, 13:3 (2008), 191  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    8. Kostko, AL, “On the bi-Hamiltonian structures for the Goryachev-Chaplygin top”, Regular & Chaotic Dynamics, 13:1 (2008), 38  mathscinet  zmath  adsnasa  isi
    9. A. V. Tsyganov, “Razdelenie peremennykh dlya odnogo obobscheniya sistemy Chaplygina na sfere”, Nelineinaya dinam., 11:1 (2015), 179–185  mathnet  elib
    10. 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
    11. Yehia H.M. Elmandouh A.A., “Integrable 2D Time-Irreversible Systems with a Cubic Second Integral”, Adv. Math. Phys., 2016, 8958747  crossref  mathscinet  zmath  isi  elib  scopus
    12. A. V. Tsiganov, “Bäcklund transformations for the Jacobi system on an ellipsoid”, Theoret. and Math. Phys., 192:3 (2017), 1350–1364  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. Elmandouh A.A., “New Integrable Problems in a Rigid Body Dynamics With Cubic Integral in Velocities”, Results Phys., 8 (2018), 559–568  crossref  isi  scopus  scopus
    14. 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
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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