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Fractional Nambu mechanics. (English) Zbl 1170.70009
Summary: The fractional generalization of Nambu mechanics is constructed by using differential forms and exterior derivatives of fractional orders. The generalized Pfaffian equations are obtained, and an example is investigated in details.
MSC:
70H99Hamiltonian and Lagrangian mechanics
70G45Differential-geometric methods for dynamical systems
26A33Fractional derivatives and integrals (real functions)
References:
[1]Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[2]Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A. Math. Gen. 39, 10375–10384 (2006) · Zbl 1097.49021 · doi:10.1088/0305-4470/39/33/008
[3]Baleanu, D.: Killing-Yano tensors and Nambu tensors. Nuovo Cim. B 114(9), 1065–1072 (1999)
[4]Baleanu, D., Muslih, S.: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 72, 119–121 (2005) · Zbl 1122.70360 · doi:10.1238/Physica.Regular.072a00119
[5]Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)
[6]Cayley, A.: Collected Mathematical Papers, vol. 3, pp. 156–204. Cambridge University Press, Cambridge (1890)
[7]Fecko, M.: On a geometric formulation of the Nambu dynamics. J. Math. Phys. 33, 926–929 (1992) · Zbl 0850.70184 · doi:10.1063/1.529744
[8]Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)
[9]Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
[10]Klimek, K.: Fractional sequential mechanics-models with symmetric fractional derivative. Czechoslov. J. Phys. E 51, 1348–1354 (2001) · Zbl 1064.70507 · doi:10.1023/A:1013378221617
[11]Klimek, K.: Lagrangean and Hamiltonian fractional sequential mechanics. Czechoslov. J. Phys. E 52, 1247–1253 (2002) · Zbl 1064.70013 · doi:10.1023/A:1021389004982
[12]Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135–3145 (2000) · doi:10.1103/PhysRevE.62.3135
[13]Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)
[14]Miller, K.S., Ross, B.: An Introduction to the Fractional Integrals and Derivatives-Theory and Application. Wiley, New York (1993)
[15]Mukund, N., Sudarshan, E.C.G.: Relation between Nambu and Hamiltonian mechanics. Phys. Rev. D 13(10), 2846–2851 (1976) · doi:10.1103/PhysRevD.13.2846
[16]Muslih, S., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. J. Math. Anal. Appl. 304, 599–606 (2005) · Zbl 1149.70320 · doi:10.1016/j.jmaa.2004.09.043
[17]Naber, M.: Time fractional Schrodinger equation. J. Math. Phys. 45, 3339–3352 (2004) · Zbl 1071.81035 · doi:10.1063/1.1769611
[18]Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2412 (1973) · doi:10.1103/PhysRevD.7.2405
[19]Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi B 133, 425–430 (1986) · doi:10.1002/pssb.2221330150
[20]Ogawa, T., Sagae, T.: Int. J. Theor. Phys. 39(12), 2875–2890 (2000) · Zbl 0993.70013 · doi:10.1023/A:1026421401600
[21]Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, New York (1974)
[22]Pandit, S.A., Gangal, A.D.: On generalized Nambu mechanics. J. Phys. A: Math. Gen. 31, 2899–2912 (1998) · Zbl 0924.70018 · doi:10.1088/0305-4470/31/12/014
[23]Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)
[24]Rabei, E.M., Nawafleh, K.I., Hiijawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007) · Zbl 1104.70012 · doi:10.1016/j.jmaa.2006.04.076
[25]Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996) · doi:10.1103/PhysRevE.53.1890
[26]Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997) · doi:10.1103/PhysRevE.55.3581
[27]Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon & Breach, New York (1993)
[28]Takhtajan, L.: On foundation of generalized Nambu mechanics. Commun. Math. Phys. 160, 295–315 (1994) · Zbl 0808.70015 · doi:10.1007/BF02103278
[29]Tarasov, V.E.: Continuous medium model for fractal media. Phys. Lett. A 336, 167–174 (2005) · Zbl 1136.81443 · doi:10.1016/j.physleta.2005.01.024
[30]Tarasov, V.E.: Fractional variation for dynamical systems: Hamilton and Lagrange approaches. J. Phys. 39(26), 8409–8425 (2006)
[31]Tarasov, V.E.: Fractional statistical mechanics. Chaos 16, 033108–033115 (2006) · Zbl 1152.82325 · doi:10.1063/1.2219701
[32]Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002) · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9
[33]Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)