The Hamilton formalism with fractional derivatives. (English) Zbl 1104.70012

Summary: Recently the traditional calculus of variations has been extended to be applicable to systems containing fractional derivatives. In this paper the passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. The Hamilton’s equations of motion are obtained in a similar manner to the usual mechanics. In addition, the classical fields with fractional derivatives are investigated using Hamiltonian formalism. Two discrete problems and one continuous are considered to demonstrate the application of the formalism, and the results obtained are in exact agreement with Agrawal formalism.


70H05 Hamilton’s equations
26A33 Fractional derivatives and integrals
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[1] Riewe, F., Non-conservative Lagrangian and Hamiltonian mechanics, Phys. rev. E, 53, 1890, (1996)
[2] Riewe, F., Mechanics with fractional derivatives, Phys. rev. E, 55, 3581, (1997)
[3] Agrawal, O.P., A new Lagrangian and a new Lagrange equation of motion for fractionally damped systems, J. appl. mech., 53, 339, (2001) · Zbl 1110.74310
[4] Rekhriashvili, S.Sh., The Lagrangian formalism with fractional derivatives in problems of mechanics, Technical phys. lett., 30, 55, (2004)
[5] Rabei, Eqab M.; Al Halholy, Tareg S.; Taani, A.A., On Hamiltonian formulation of non-conservative systems, Turkish J. phys., 28, 4, 213, (2004)
[6] Rabei, Eqab M.; Al Halholy, Tareg S.; Rousan, A., Potentials of arbitrary forces with fractional derivatives, Internat. J. modern phys. A, 19, 3083, (2004) · Zbl 1080.70516
[7] Rousan, A.; Malkawi, E.; Rabei, Eqab M.; Widyan, H., Applications of fractional calculus of gravity, Fract. calc. appl. anal., 5, 155, (2002) · Zbl 1039.86007
[8] O.P. Agrawal, An Analytical Scheme for Stochastic Dynamics Systems Containing Fractional Derivatives, ASME Design Engineering Technical Conferences, 1999
[9] Agrawal, O.P., Formulation of euler – lagrange equations for variational problems, J. math. anal. appl., 272, 368, (2002) · Zbl 1070.49013
[10] Baleanu, D.; Muslih, S., Lagrangian formulation of classical fields within riemann – liouville fractional derivatives, Phys. scripta, 27, 105, (2005) · Zbl 1122.70360
[11] Dreisi-gmeyer, David W.; Yoang, P.M., Non-conservative Lagrangian mechanics: A generalized functional approach, J. phys. A, 36, 3297, (2003)
[12] Klimek, M., Lagrangian and Hamiltonian fractional sequential mechanics, Czechoslovak J. phys., 52, 1247, (2002) · Zbl 1064.70013
[13] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[14] Goldstein, H., Classical mechanics, (1980), Addison Wesley · Zbl 0491.70001
[15] Muslih, S.; Baleanu, D., Hamiltonian formulation of systems with linear velocities within riemann – liouville fractional derivatives, J. math. anal. appl., 304, 599, (2005) · Zbl 1149.70320
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