Fractional Hamilton’s equations of motion in fractional time

• Autorzy: Sami Muslih , Dumitru Baleanu , Eqab Rabei
• Języki publikacji: EN

Abstrakty

EN

The Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton’s equations are obtained and two examples are investigated in detail.

Słowa kluczowe

EN

Fractional calculus; fractional derivatives; fractional Lagrangian and Hamiltonian; fractional time

• Czasopismo: Open Physics
• Rocznik: 2007
• Tom: 5
• Numer: 4
• Strony: 549-557

Daty

wydano

2007-12-01

online

2007-12-01

Twórcy

autor

Sami Muslih,

autor

Dumitru Baleanu

• Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530, Ankara, Turkey,

autor

Eqab Rabei

• Department of Physics, Mutah University, 1324, Karak, Jordan

Bibliografia

• [1] X. He: “Anisrtopy and isotropy: a model of fraction-dimensional space”, Solid State Comm., Vol. 75, (1990), pp. 111–114. http://dx.doi.org/10.1016/0038-1098(90)90352-C[Crossref]
• [2] K.G. Willson: “Quantum field-theory, models in less than 4 dimensions”, Phys. Rev. D, Vol. 7, (1973), pp. 2911–2926. http://dx.doi.org/10.1103/PhysRevD.7.2911
• [3] F.H. Stillinger: “Axiomatic basis for spaces with non-integer dimensions”, J. Math. Phys., Vol. 18, (1977), pp. 1224–1234. http://dx.doi.org/10.1063/1.523395[Crossref]
• [4] A. Zeilinger and K. Svozil: “Measuring the dimension of space time”, Phys. Rev. Lett., Vol. 54, (1995), pp. 2553–2555. http://dx.doi.org/10.1103/PhysRevLett.54.2553[Crossref]
• [5] M.A. Lohe and A. Thilagam: “Quantum mechanical models in fractional dimesions”, J. Phys. A, Vol. 37, (2004), pp. 6181–6199. http://dx.doi.org/10.1088/0305-4470/37/23/015[Crossref]
• [6] C. Palmer and P.N. Stavrinou: “Equations of motion in a non-integer-dimensional space”, J. Phys. A, Vol. 37, (2004) pp. 6986–7003. http://dx.doi.org/10.1088/0305-4470/37/27/009[Crossref]
• [7] C.M. Bender and K.A. Milton: “Scalar Casimir effect for a D-dimensional sphere”, Phys. Rev. D, Vol. 50, (1994), pp. 6547–7555. http://dx.doi.org/10.1103/PhysRevD.50.6547[Crossref]
• [8] C.W. Misner, K.S. Thorne and J.A. Wheeler: Gravitation, Freeman, San Francisco, 1975.
• [9] A. Zeilinger and K. Svozil: “Measuring the dimension of space time”, Phys. Rev. Lett., Vol. 54, (1995), pp. 2553–2555. http://dx.doi.org/10.1103/PhysRevLett.54.2553[Crossref]
• [10] K. Svozil: “Quantum field theory on fractal spacetime: a new regularisation method”, J. Phys. A., Vol. 20, (1987), pp. 3861–3875. http://dx.doi.org/10.1088/0305-4470/20/12/033[Crossref]
• [11] F.Y. Ren, J.R. Liang, X.T. Wang and W.Y. Qiu: “Integrals and derivatives on net fractals”, Chaos, Soliton and Fractals, Vol. 16, (2003), pp. 107–117. http://dx.doi.org/10.1016/S0960-0779(02)00211-4[Crossref]
• [12] K.S. Miller and B. Ross: An Introduction to the Fractional Calculus and Fractional Differential Equations., John Wiley and Sons Inc., New York, 1993.
• [13] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Linghorne, P.A., 1993.
• [14] K.B. Oldham and J. Spanier: The Fractional Calculus, Academic Press, New York, 1974.
• [15] I. Podlubny: Fractional Differential Equations, Academic Press, New York, 1999.
• [16] A.A. Kilbas, H.H. Srivastava and J.J. Trujillo: Theory and Applications of Fractional Differential Equations, Elsevier, (2006).
• [17] R. Gorenflo and F. Mainardi: Fractional calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continoum Mechanics, Springer Verlag, Wien and New York, 1997.
• [18] G.M. Zaslavsky: “Chaos, fractional kinetics, and anomalous transport”, Phys. Rep., Vol. 371, (2002), pp. 461–580. http://dx.doi.org/10.1016/S0370-1573(02)00331-9[Crossref]
• [19] F. Mainardi: “Fractional relaxation-oscillation and fractional diffusion-wave phenomena”, Chaos, Solitons and Fractals, Vol. 7, (1996), pp. 1461–1477. http://dx.doi.org/10.1016/0960-0779(95)00125-5[Crossref]
• [20] E. Scalas, R. Gorenflo and F. Mainardi: “Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation”, Phys. Rev. E, Vol. 69, (2004), art. 011107.
• [21] F. Mainardi, G. Pagnini and R. Gorenflo: “Mellin transform and subordination laws in fractional diffusion processes”, Frac. Calc. Appl. Anal., Vol. 6, (2003), pp. 441–459.
• [22] J.A. Tenreiro-Machado: “Discrete-time Fractional-order controllers”, Frac. Calc. Appl. Anal., Vol. 4, (2001), pp. 47–68.
• [23] F. Riewe: “Nonconservative Lagrangian and Hamiltonian mechanics”, Phys. Rev. E, Vol. 53, (1996), pp. 1890–1899. http://dx.doi.org/10.1103/PhysRevE.53.1890[Crossref]
• [24] F. Riewe: “Mechanics with fractional derivatives”, Phys. Rev. E, Vol. 55, (1997), pp. 3581–3592. http://dx.doi.org/10.1103/PhysRevE.55.3581[Crossref]
• [25] O.P. Agrawal: “Formulation of Euler-Lagrange equations for fractional variational problems”, J. Math. Anal. Appl., Vol. 272, (2002), pp. 368–379. http://dx.doi.org/10.1016/S0022-247X(02)00180-4[Crossref]
• [26] M. Klimek: “Fractional sequential mechanics-models with symmetric fractional derivatives”, Czech. J. Phys., Vol. 51, (2001), pp. 1348–1354. http://dx.doi.org/10.1023/A:1013378221617[Crossref]
• [27] M. Klimek: “Lagrangian and Hamiltonian fractional seqential mechanics”, Czech. J. Phys., Vol. 52, (2002), pp. 1247–1253. http://dx.doi.org/10.1023/A:1021389004982[Crossref]
• [28] M. Klimek: “Stationarity-conservation laws for certain linear fractional differential equations”, J. Phys. A-Math. Gen., Vol. 34, (2001), pp. 6167–6184. http://dx.doi.org/10.1088/0305-4470/34/31/311[Crossref]
• [29] A. Raspini: “Simple Solutions of the Fractional Dirac Equation of Order 2/3”, Physica Scripta, Vol. 64, (2001), pp. 20–22. http://dx.doi.org/10.1238/Physica.Regular.064a00020[Crossref]
• [30] M. Naber: “Time fractional Schrödinger equation”, J. Math. Phys., Vol. 45, (2004), pp. 3339–3352. http://dx.doi.org/10.1063/1.1769611[Crossref]
• [31] R.A. El-Nabulsi: “A fractional approach to nonconservative Lagrangian dynamics”, Fizika A, Vol. 14, (2005), pp. 289–298.
• [32] S.I. Muslih, D. Baleanu and E. Rabei: “Hamiltonian formulation of classical fields within Riemann-Liouville fractional derivatives”, Physica Scripta, Vol. 73, (2006), pp. 436–438. http://dx.doi.org/10.1088/0031-8949/73/5/003[Crossref]
• [33] D. Baleanu and T. Avkar: “Lagrangians with linear velocities within Riemann-Liouville fractional derivatives”, Nuovo Cimento, Vol. 119, (2004), pp. 73–79.
• [34] S. Muslih and D. Baleanu: “Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives”, J. Math. Anal. Appl., Vol. 304, (2005), pp. 599–603. http://dx.doi.org/10.1016/j.jmaa.2004.09.043[Crossref]
• [35] D. Baleanu and S. Muslih: “Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives”, Physica Scripta, Vol. 72, (2005), pp. 119–121. http://dx.doi.org/10.1238/Physica.Regular.072a00119[Crossref]
• [36] D. Baleanu and O.P. Agrawal: “Fractional Hamilton formalism within Caputo’s derivative”, Czech. J. Phys.,(2006), Vol. 56, pp. 1087–1092. http://dx.doi.org/10.1007/s10582-006-0406-x[Crossref]
• [37] D. Baleanu and S.I. Muslih: “About fractional supersymmetric quantum mechanics”, Czech. J. Phys., Vol. 55, (2005), pp. 1063–1066. http://dx.doi.org/10.1007/s10582-005-0106-y[Crossref]
• [38] D. Baleanu and S.I. Muslih: “Formulation of Hamiltonian equations for fractional variational problems”, Czech. J. Phys., Vol. 55, (2005), pp. 633–642. http://dx.doi.org/10.1007/s10582-005-0067-1[Crossref]
• [39] A.A. Stanislavsky: “Hamiltonian formalism of fractional systems”, Eur. Phys. J. B, Vol. 49, (2006), pp. 93–101. http://dx.doi.org/10.1140/epjb/e2006-00023-3[Crossref]
• [40] E.M. Rabei, K.I. Nawafleh, R.S. Hijjawi, S.I. Muslih and D. Baleanu: “The Hamilton formalism with fractional derivatives”, J. Math. Anal. Appl., Vol. 327, (2007), pp. 891–897. http://dx.doi.org/10.1016/j.jmaa.2006.04.076[Crossref]
• [41] G.S.F. Fredericoa and F.M. Torres: “A formulation of Noether’s theorem for fractional problems of the calculus of variations”, J. Math. Anal. Appl., in press, 2007. [Crossref][WoS]
• [42] O.P. Agrawal and D. Baleanu: “A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems”, J. Vibr. Contr., in press, 2007. [Crossref]
• [43] A.A. Stanislavsky: “Probability interpretation of the integral of fractional order”, Theor. Math. Phys., Vol. 138, (2004), pp. 418–431. http://dx.doi.org/10.1023/B:TAMP.0000018457.70786.36[Crossref]
• [44] R.R. Nigmatullin: “The fractional integral and its physical interpretation”, Theor. Math. Phys., Vol. 90, (1992), pp. 242–251. http://dx.doi.org/10.1007/BF01036529[Crossref]
• [45] V.E. Tarasov: “Electromagnetic fields on fractals”, Mod. Phys. Lett. A, Vol. 12, (2006), pp. 1587–1600. http://dx.doi.org/10.1142/S0217732306020974[Crossref]
• [46] G. Jumarie: “Lagrangian mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor’s series of nondifferntiable functions”, Chaos, Solitons and Fractals Vol. 32, (2007), pp. 969–987. http://dx.doi.org/10.1016/j.chaos.2006.07.053[Crossref][WoS]
• [47] R.A. El-Nabulsi: “Differential Geometry and Modern Cosmology with Fractionaly Differentiated Lagrangian Function and Fractional Decaying Force Term”, Rom. J. Phys., Vol. 52, (2007), pp. 441–450.

Identyfikatory

DOI

10.2478/s11534-007-0041-6