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TMF, 2007, Volume 151, Number 1, Pages 26–43 (Mi tmf6009)  

This article is cited in 7 scientific papers (total in 7 papers)

Compatible Lie–Poisson brackets on the Lie algebras $e(3)$ and $so(4)$

A. V. Tsiganov

St. Petersburg State University, Faculty of Physics

Abstract: We completely classify the compatible Lie–Poisson brackets on the dual spaces of the Lie algebras $e(3)$ and $so(4)$. The corresponding bi-Hamiltonian systems are the spinning tops corresponding to the classical cases of integrability of the Euler equations, the Kirchhoff equations, and the Poincaré–Zhukovskii equations.

Keywords: integrable system, bi-Hamiltonian manifold, Lie–Poisson tensor

DOI: https://doi.org/10.4213/tmf6009

Full text: PDF file (493 kB)
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English version:
Theoretical and Mathematical Physics, 2007, 151:1, 459–473

Bibliographic databases:

Received: 03.07.2006
Revised: 12.10.2006

Citation: A. V. Tsiganov, “Compatible Lie–Poisson brackets on the Lie algebras $e(3)$ and $so(4)$”, TMF, 151:1 (2007), 26–43; Theoret. and Math. Phys., 151:1 (2007), 459–473

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/tmf/v151/i1/p26

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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. Tsiganov, AV, “Separation of variables for a pair of integrable systems on so*(4)”, Doklady Mathematics, 76:3 (2007), 839  crossref  mathscinet  zmath  isi  elib  scopus
    2. Tsiganov, AV, “A family of the Poisson brackets compatible with the Sklyanin bracket”, Journal of Physics A-Mathematical and Theoretical, 40:18 (2007), 4803  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Tsiganov AV, “The Poisson bracket compatible with the classical reflection equation algebra”, Regular & Chaotic Dynamics, 13:3 (2008), 191–203  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Tsiganov AV, “On bi-Hamiltonian structure of some integrable systems on so*(4)”, Journal of Nonlinear Mathematical Physics, 15:2 (2008), 171–185  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. Tsiganov A.V., “New Variables of Separation for Particular Case of the Kowalevski Top”, Regular & Chaotic Dynamics, 15:6 (2010), 659–669  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. A. V. Tsyganov, “O novom razdelenii peremennykh dlya chastnogo sluchaya volchka Kovalevskoi”, Nelineinaya dinam., 6:3 (2010), 639–652  mathnet
    7. Alina Dobrogowska, Anatol Odzijewicz, “Integrable Systems Related to Deformed $\mathfrak{so}(5)$”, SIGMA, 10 (2014), 056, 18 pp.  mathnet  crossref  mathscinet
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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