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SIGMA 10 (2014), 056, 18 pages arXiv:1311.0679
https://doi.org/10.3842/SIGMA.2014.056
Integrable Systems Related to Deformed $\mathfrak{so}(5)$
Alina Dobrogowska and Anatol Odzijewicz
Institute of Mathematics, University of Białystok, Lipowa 41, 15-424 Białystok, Poland
Received November 05, 2013, in final form May 26, 2014; Published online June 03, 2014
Abstract
We investigate a family of integrable Hamiltonian systems on Lie-Poisson spaces $\mathcal{L}_+(5)$ dual to Lie algebras $\mathfrak{so}_{\lambda, \alpha}(5)$ being two-parameter deformations of $\mathfrak{so}(5)$. We integrate corresponding Hamiltonian equations on $\mathcal{L}_+(5)$ and $T^*\mathbb{R}^5$ by quadratures as well as discuss their possible physical interpretation.
Key words:
integrable Hamiltonian systems; Casimir functions; Lie algebra deformation; symplectic dual pair; momentum map.
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References
-
Adler M., van Moerbeke P., Vanhaecke P., Algebraic integrability, Painlevé geometry and Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 47, Springer-Verlag, Berlin, 2004.
-
Bolsinov A.V., Borisov A.V., Compatible Poisson brackets on Lie algebras, Math. Notes 72 (2002), 10-30.
-
Borisov A.V., Mamaev I.S., Poisson structures and Lie algebras in Hamiltonian mechanics, Udmurtskii Universitet, Izhevsk, 1999.
-
Butcher P.N., Cotter D., The elements of nonlinear optics, Cambridge University Press, Cambridge, 1990.
-
Cannas da Silva A., Weinstein A., Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, Vol. 10, Amer. Math. Soc., Providence, RI, 1999.
-
Dobrogowska A., Odzijewicz A., Integrable relativistic systems given by Hamiltonians with momentum-spin-orbit coupling, Regul. Chaotic Dyn. 17 (2012), 492-505, arXiv:1106.4480.
-
Dobrogowska A., Ratiu T.S., Integrable systems of Neumann type, J. Dynam. Differential Equations, to appear.
-
Griffiths P., Harris J., Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience, New York, 1978.
-
Holm D.D., Geometric mechanics. Part I. Dynamics and symmetry, 2nd ed., Imperial College Press, London, 2011.
-
Komarov I.V., Sokolov V.V., Tsiganov A.V., Poisson maps and integrable deformations of the Kowalevski top, J. Phys. A: Math. Gen. 36 (2003), 8035-8048, nlin.SI/0304033.
-
Libermann P., Marle C.-M., Symplectic geometry and analytical mechanics, Mathematics and its Applications, Vol. 35, D. Reidel Publishing Co., Dordrecht, 1987.
-
Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
-
Magri F., A geometrical approach to the nonlinear solvable equations, in Nonlinear Evolution Equations and Dynamical Systems (Lecce, 1979), Lecture Notes in Phys., Vol. 120, Springer, Berlin - New York, 1980, 233-263.
-
Mishchenko A.S., Vector bundles and their applications, Nauka, Moscow, 1984.
-
Odzijewicz A., Dobrogowska A., Integrable Hamiltonian systems related to the Hilbert-Schmidt ideal, J. Geom. Phys. 61 (2011), 1426-1445, arXiv:1004.3955.
-
Odzijewicz A., Goliński T., Hierarchy of integrable Hamiltonians describing the nonlinear $n$-wave interaction, J. Phys. A: Math. Theor. 45 (2012), 045204, 14 pages, arXiv:1106.3217.
-
Perelomov A.M., Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990.
-
Sokolov V.V., Wolf T., Integrable quadratic classical Hamiltonians on ${\rm so}(4)$ and ${\rm so}(3,1)$, J. Phys. A: Math. Gen. 39 (2006), 1915-1926, nlin.SI/0405066.
-
Trofimov V.V., Fomenko A.T., Algebra and geometry of integrable Hamiltonian differential equations, Faktorial, Moscow, 1995.
-
Tsiganov A.V., Compatible Lie-Poisson brackets on the Lie algebras ${\rm e}(3)$ and ${\rm so}(4)$, Theoret. and Math. Phys. 151 (2007), 459-473.
-
Tsiganov A.V., On bi-Hamiltonian structure of some integrable systems on ${\rm so}^*(4)$, J. Nonlinear Math. Phys. 15 (2008), 171-185, nlin.SI/0703062.
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