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

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

Francisco Crespo 1, , Francisco Javier Molero 2, and  Sebastián Ferrer 2,

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.
Keywords: Poisson bracket, Differential system, Nambu dynamics, Jacobian multiplier., completely integrable.
Mathematics Subject Classification: Primary: 34A26, 34A34; Secondary: 34A30, 70H0.
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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

show all references

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

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