Abstract
The paper presents fractional Hamilton–Jacobi formulations for systems containing Riesz fractional derivatives (RFD’s). The Hamilton–Jacobi equations of motion are obtained. An illustrative example for simple harmonic oscillator (SHO) has been discussed. It was observed that the classical results are recovered for integer order derivatives.
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References
Agrawal, O.P.: Nonlinear Dyn. 38, 323 (2004)
Agrawal, O.P.: Generalized Euler–Lagrange equations and transversality conditions for FVPs in terms of Caputo derivative. In: Tas, K., Tenreiro-Machado, J.A., Baleanu, D. (eds.) Proc. MME06, Ankara, Turkey (2006)
Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A, Math. Theor. 40, 6287–6303 (2007)
Agrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. In: Tas, K., Tenreiro-Machado, J.A., Baleanu, D. (eds.) Proc. MME06, Ankara, Turkey (2006)
Agrawal, O.P., Tenreiro-Machado, J.A., Sabatier, J. (eds.): Fractional Derivatives and Their Application (2004)
Baleanu, D.: Constrained systems and Riemann–Liouville fractional derivatives. In: Fractional Differentiation and Its Applications. U-Books Verlag, Augsburg (2005)
Baleanu, D.: Signal Process. 86(10), 2632 (2006). Dynamics. Springer, Berlin (2006)
Baleanu, D.: New applications of fractional variational principles. Rep. Math. Phys. 61(2), 199–206 (2008)
Baleanu, D., Muslih, S.: Phys. Scr. 72(2–3), 119 (2005)
Baleanu, D., Muslih, S.: New trends in fractional quantization method. In: Intelligent Systems at the Service of Mankind. 1st edn., vol. 2, U-Books Verlag, Augsburg (2005)
Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, Berlin (1997)
Hilfer, E. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publisher, Redding (2006)
Metzler, R., Klafter, J.: Phys. Rep. 339, 1–7 (2000)
Metzler, R., Klafter, J.: J. Phys. A, Math. Gen. 37, R161–208 (2004). Mod. Phys. A 19, 3083 (2004)
Momani, S., Qaralleh, R.: An efficient method for solving systems of fractional integro-differential equations. Comput. Math. Appl. 52, 459–470 (2006). Nonlinear Dynamics, vol. 38. Springer, Berlin (2006)
Muslih, S.I., Baleanu, D.: J. Math. Anal. Appl. 304, 599 (2005)
Muslih, S., Baleanu, D.: Czechoslov. J. Phys. 55(6), 633 (2005)
Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)
Rabei, E.M.: On Hamiltonian systems with constraints. Hadron. J. 19, 597 (1996)
Rabei, E.M., Güler, Y.: The Hamilton–Jacobi treatment of second class constraints. Phys. Rev. A 46, 3513 (1992)
Rabei, E.M., Ajlouni, A.W., Ghassib, H.B.: Int. J. Theor. Phys. 45, 1619 (2006)
Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamiltonian formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007)
Tenreiro-Machado, J.A.: Special Issue of Fractional Order Calculus and Its Applications, Nonlinear(ed) (2002)
Tenreiro-Machado, J., Kiryakova, V., Mainardi, F.: Commun. Nonlinear Sci. Numer. Simul. 16, 1140–1153 (2011)
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Rabei, E.M., Rawashdeh, I.M., Muslih, S. et al. Hamilton–Jacobi Formulation for Systems in Terms of Riesz’s Fractional Derivatives. Int J Theor Phys 50, 1569–1576 (2011). https://doi.org/10.1007/s10773-011-0668-3
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DOI: https://doi.org/10.1007/s10773-011-0668-3