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Number of results

Journal

2007 | 5 | 4 | 549-557

Article title

Fractional Hamilton’s equations of motion in fractional time

Content

Title variants

Languages of publication

EN

Abstracts

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.

Publisher

Journal

Year

Volume

5

Issue

4

Pages

549-557

Physical description

Dates

published
1 - 12 - 2007
online
1 - 12 - 2007

Contributors

author
  • Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530, Ankara, Turkey
author
  • Department of Physics, Mutah University, 1324, Karak, Jordan

References

  • [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.

Document Type

Publication order reference

Identifiers

YADDA identifier

bwmeta1.element.-psjd-doi-10_2478_s11534-007-0041-6
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