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Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

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  • Poisson and integrable systems are orbitally equivalent through the Nambu bracket. Namely, we show that every completely integrable system of differential equations may be expressed into the Poisson-Hamiltonian formalism by means of the Nambu-Hamilton equations of motion and a reparametrisation related by the Jacobian multiplier. The equations of motion provide a natural way for finding the Jacobian multiplier. As a consequence, we partially give an alternative proof of a recent theorem in [13]. We complete this work presenting some features associated to Hamiltonian maximally superintegrable systems.
    Mathematics Subject Classification: Primary: 34A26, 34A34; Secondary: 34A30, 70H05.

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  • [1]

    F. Bayen and M. Flato, Remarks concerning Nambu's generalized mechanics, Phys. Rev. D, 11 (1975), 3049-3053.doi: 10.1103/PhysRevD.11.3049.

    [2]

    R. Chatterjee, Dynamical symmetries and Nambu mechanics, Letters in Mathematical Physics, 36 (1996), 117-126.doi: 10.1007/BF00714375.

    [3]

    R. Chatterjee and L. Takhtajan, Aspects of classical and quantum Nambu mechanics, Letters in Mathematical Physics, 37 (1996), 475-482.doi: 10.1007/BF00312678.

    [4]

    S. Codriansky, R. Navarro and M. Pedroza, The Liouville condition and Nambu mechanics, Journal of Physics A: Mathematical and General, 29 (1996), 1037-1044.doi: 10.1088/0305-4470/29/5/017.

    [5]

    I. Cohen and A. Kálnay, On Nambu's generalized Hamiltonian mechanics, International Journal of Theoretical Physics, 12 (1975), 61-67.doi: 10.1007/BF01884111.

    [6]

    F. Crespo and S. Ferrer, On the extended Euler system and the Jacobi and Weierstrass elliptic functions, Journal of Geometric Mechanics, 7 (2015), 151-168.

    [7]

    F. Crespo and S. Ferrer, On the Nambu-Poisson systems and its integrability, In preparation, 2015b.

    [8]

    S. Ferrer, F. Crespo and F. J. Molero, On the N-Extended Euler system I. Generalized Jacobi elliptic functions, Nonlinear Dynamics, 84 (2016), 413-435.

    [9]

    Gautheron, P., Some remarks concerning Nambu mechanics, Letters in Mathematical Physics, 37 (1996), 103-116.doi: 10.1007/BF00400143.

    [10]

    A. Horikoshi and Y. Kawamura, Hidden Nambu mechanics: A variant formulation of hamiltonian systems, Progress of Theoretical and Experimental Physics, 2013.

    [11]

    R. Ibáñez, M. de León, J. Marrero and D. Martín de Diego, Dynamics of generalized Poisson and NambuPoisson brackets, Journal of Mathematical Physics, 38 (1997), 2332-2344.doi: 10.1063/1.531960.

    [12]

    R. Ibáñez, M. de León, J. Marrero and D. Martín de Diego, Reduction of generalized Poisson and Nambu-Poisson manifolds, Reports on Mathematical Physics, 42 (1998), 71-90. Proceedings of the Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraints in Dynamics.doi: 10.1016/S0034-4877(98)80005-0.

    [13]

    J. Llibre, C. Valls and X. Zhang, The completely integrable differential systems are essentially linear differential systems, Journal of Nonlinear Science, 25 (2015), 815-826.doi: 10.1007/s00332-015-9243-z.

    [14]

    N. Makhaldiani, Nambu-Poisson dynamics with some applications, Physics of Particles and Nuclei, 43 (2012), 703-707.

    [15]

    J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag New York, Inc., 2nd edition, 1999.

    [16]

    K. Modin, Time transformation and reversibility of Nambu-Poisson systems, J. Gen. Lie Theory Appl., 3 (2009), 39-52.doi: 10.4303/jglta/S080103.

    [17]

    P. Morando, Liouville condition, Nambu mechanics, and differential forms, Journal of Physics A: Mathematical and General, 29 (1996), L329-L331.doi: 10.1088/0305-4470/29/13/004.

    [18]

    N. Mukunda and E. C. G. Sudarshan, Relation between Nambu and Hamiltonian mechanics, Phys. Rev. D, 13 (1976), 2846-2850.doi: 10.1103/PhysRevD.13.2846.

    [19]

    Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev., 7 (1973), 2405-2412.doi: 10.1103/PhysRevD.7.2405.

    [20]

    S. Pandit and A. Gangal, On generalized Nambu mechanics, Journal of Physics A: Mathematical and General, 31 (1998), 2899-2912.doi: 10.1088/0305-4470/31/12/014.

    [21]

    O. Rössler, An equation for continuous chaos, Phys. Lett. A, 57 (1987), 397-398.

    [22]

    L. Takhtajan, On foundation of the generalized Nambu mechanics, Communications in Mathematical Physics, 160 (1994), 295-315.doi: 10.1007/BF02103278.

    [23]

    R. Tudoran and A. Gîrban, On the completely integrable case of the Rössler system, Journal of Mathematical Physics, 2012.

    [24]

    I. Vaisman, A survey on Nambu-Poisson brackets, Acta Mathematica Universitatis Comenianae. New Series, 68 (1999), 213-241.

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