# American Institute of Mathematical Sciences

June  2016, 8(2): 169-178. doi: 10.3934/jgm.2016002

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

 1 Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C. Concepción, Chile 2 Departamento de Matemática Aplicada, Universidad de Murcia, 30071 Espinardo, Spain

Received  August 2015 Revised  January 2016 Published  June 2016

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.
Citation: Francisco Crespo, Francisco Javier Molero, Sebastián Ferrer. Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier. Journal of Geometric Mechanics, 2016, 8 (2) : 169-178. doi: 10.3934/jgm.2016002
##### References:
 [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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##### References:
 [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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