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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 |
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. Google Scholar |
[7] |
F. Crespo and S. Ferrer, On the Nambu-Poisson systems and its integrability,, In preparation, (2015). 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. Google Scholar |
[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). 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.
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. Google Scholar |
[15] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag New York, (1999). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[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. Google Scholar |
[7] |
F. Crespo and S. Ferrer, On the Nambu-Poisson systems and its integrability,, In preparation, (2015). 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. Google Scholar |
[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). 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.
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. Google Scholar |
[15] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry,, Springer-Verlag New York, (1999). Google Scholar |
[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. Google Scholar |
[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). Google Scholar |
[24] |
I. Vaisman, A survey on Nambu-Poisson brackets,, Acta Mathematica Universitatis Comenianae. New Series, 68 (1999), 213.
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