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

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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

Abstract
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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.

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. doi: 10.1103/PhysRevD.11.3049.
[2]	R. Chatterjee, Dynamical symmetries and Nambu mechanics,, Letters in Mathematical Physics, 36 (1996), 117. doi: 10.1007/BF00714375.
[3]	R. Chatterjee and L. Takhtajan, Aspects of classical and quantum Nambu mechanics,, Letters in Mathematical Physics, 37 (1996), 475. 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. 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. 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.
[7]	F. Crespo and S. Ferrer, On the Nambu-Poisson systems and its integrability,, In preparation, (2015).
[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.
[9]	Gautheron, P., Some remarks concerning Nambu mechanics,, Letters in Mathematical Physics, 37 (1996), 103. 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. 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. 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. doi: 10.1007/s00332-015-9243-z.
[14]	N. Makhaldiani, Nambu-Poisson dynamics with some applications,, Physics of Particles and Nuclei, 43 (2012), 703.
[15]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag New York, (1999).
[16]	K. Modin, Time transformation and reversibility of Nambu-Poisson systems,, J. Gen. Lie Theory Appl., 3 (2009), 39. doi: 10.4303/jglta/S080103.
[17]	P. Morando, Liouville condition, Nambu mechanics, and differential forms,, Journal of Physics A: Mathematical and General, 29 (1996). 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. doi: 10.1103/PhysRevD.13.2846.
[19]	Y. Nambu, Generalized Hamiltonian mechanics,, Phys. Rev., 7 (1973), 2405. 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. doi: 10.1088/0305-4470/31/12/014.
[21]	O. Rössler, An equation for continuous chaos,, Phys. Lett. A, 57 (1987), 397.
[22]	L. Takhtajan, On foundation of the generalized Nambu mechanics,, Communications in Mathematical Physics, 160 (1994), 295. 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.

show all references

References:

[1]	F. Bayen and M. Flato, Remarks concerning Nambu's generalized mechanics,, Phys. Rev. D, 11 (1975), 3049. doi: 10.1103/PhysRevD.11.3049.
[2]	R. Chatterjee, Dynamical symmetries and Nambu mechanics,, Letters in Mathematical Physics, 36 (1996), 117. doi: 10.1007/BF00714375.
[3]	R. Chatterjee and L. Takhtajan, Aspects of classical and quantum Nambu mechanics,, Letters in Mathematical Physics, 37 (1996), 475. 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. 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. 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.
[7]	F. Crespo and S. Ferrer, On the Nambu-Poisson systems and its integrability,, In preparation, (2015).
[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.
[9]	Gautheron, P., Some remarks concerning Nambu mechanics,, Letters in Mathematical Physics, 37 (1996), 103. 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. 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. 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. doi: 10.1007/s00332-015-9243-z.
[14]	N. Makhaldiani, Nambu-Poisson dynamics with some applications,, Physics of Particles and Nuclei, 43 (2012), 703.
[15]	J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag New York, (1999).
[16]	K. Modin, Time transformation and reversibility of Nambu-Poisson systems,, J. Gen. Lie Theory Appl., 3 (2009), 39. doi: 10.4303/jglta/S080103.
[17]	P. Morando, Liouville condition, Nambu mechanics, and differential forms,, Journal of Physics A: Mathematical and General, 29 (1996). 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. doi: 10.1103/PhysRevD.13.2846.
[19]	Y. Nambu, Generalized Hamiltonian mechanics,, Phys. Rev., 7 (1973), 2405. 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. doi: 10.1088/0305-4470/31/12/014.
[21]	O. Rössler, An equation for continuous chaos,, Phys. Lett. A, 57 (1987), 397.
[22]	L. Takhtajan, On foundation of the generalized Nambu mechanics,, Communications in Mathematical Physics, 160 (1994), 295. 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.

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