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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), 3049-3053.
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. |
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. |
[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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