Fractional variational principles have gained considerable importance during the last decade due to their various applications in several areas of science and engineering. In this study, the fractional Euler— Lagrange equations corresponding to a prescribed fractional space are obtained. These equations are obtained using the traditional method of calculus of variations adapted to the case of fractional space. The most general fractional Lagrangian is considered and the limit case when the parameters involved in fractional derivatives are equal to one, is obtained. Two examples are investigated in this study, namely the free particle on fractional space and the fractional simple pendulum, and their corresponding fractional Euler—Lagrange equations ar obtained.
Bender, C.M. and Milton, K.A., 1994, “Scalar Casimir effect for a D-dimensional sphere ,” Physical ReviewD50(10), 6547—7555.
2.
Gorenflo, R. and Mainardi, F., 1997, Fractional calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics , Springer Verlag, Wien and NewYork.
3.
He, X., 1990, “Anisotropy and isotropy: A model of fraction-dimensional space,” Solid State Communications75(3), 111—114.
4.
Hilfer, R., 2000, Applications of Fraction Calculus in Physics, World Scientific, Singapore.
5.
Lohe, M.A. and Thilagam, A., 2004, “Quantum mechanical models in fractional dimesions ,” Journal of PhysicsA37(23), 6181—6199.
6.
Miller, K.S. and Ross, B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
7.
Misner, C.W., Thorne, K.S., and Wheeler, J.A., 1975, Gravitation, Freeman, San Francisco, CA.
8.
Oldham, K.B. and Spanier, J., 1974, The Fractional Calculus, Academic Press , New York.
9.
Palmer, C. and Stavrinou, P.N., 2004, “Equations of motion in a non-integer-dimensional space,” Journal Physics A 37(27), 6986—7003.
10.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, New York.
11.
Samko, S.G., Kilbas, A.A., and Marichev, O.I., 1993, Fractional Integrals and Derivatives — Theory and Applications, Gordon and Breach, Linghorne.
12.
Stillinger, F.H., 1977, “Axiomatic basis for spaces with non-integer dimensions,” Journal of MathematicalPhysics18(2), 1224—1234.
13.
Willson, K.G., 1973, “Quantum field-theory, models in less than 4 dimensions ” Physical ReviewD7(10), 2911— 2926.
14.
Zeilinger, A. and Svozil, K., 1995, “Measuring the dimension of space time,” Physical Review Letters54(24), 2553— 2555.
15.
Agrawal, O.P., 2002, “Formulation of Euler—Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications272(1), 368—379.
16.
Baleanu, D., 2004a, “Constrained systems and Riemann—Liouville fractional derivative,” in Proceedings of 1st IFAC Workshop on Fractional Differentiation and its Applications, July , Bordeaux, France, pp. 597—602.
17.
Baleanu, D., 2004b, “About fractional calculus of singular Lagrangians ,” in Proceedings of IEEE International Conference on Computational Cybernetics ICCC, August, Wien, Austria, pp. 379—385.
18.
Baleanu, D., 2005, “Constrained systems and Riemann—Liouville fractional derivatives”, in Fractional Differentiation and its Applications, U-Books Verlag, Augsburg, Germany.
19.
Baleanu, D. and Avkar, T., 2004, “Lagrangians with linear velocities within Riemann—Liouville fractional derivatives,” Nuovo Cimento119(1), 73—79.
20.
Baleanu, D. and Muslih, S., 2005a, “Lagrangian formulation of classical fields within Riemann—Liouville fractional derivatives,” Physica Scripta72(2—3), 119—121.
21.
Baleanu, D. and Muslih, S., 2005b, “New trends in fractional quantization method,” in Intelligent Systems at the Service of Mankind, Vol. 2, 1st edn. U-Books Verlag, Augsburg, Germany.
22.
Baleanu, D. and Muslih, S.I., 2005c, “About fractional supersymmetric quantum mechanics,” Czechoslovak Journal ofPhysics55(9), 1063—1066.
23.
Baleanu, D. and Muslih, S.I., 2005d, “Formulation of Hamiltonian equations for fractional variational problems,” Czechoslovak Journal ofPhysics55(6), 633—642.
24.
Cottrill-Shepherd, K. and Naber, M., 2001, “Fractional differential forms,” Journal of Mathematical Physics42(5), 2203—2212.
25.
Klimek, M., 2001, “Fractional sequential mechanics-models with symmetric fractional derivatives” Czechoslovak Journal of Physics51(12), 1348—1354.
26.
Klimek, M., 2002, “Lagrangian and Hamiltonian fractional seqential mechanics” Czechoslovak Journal of Physics52(11), 1247—1253.
Laskin, N., 2002b, “Fractals and quantum mechanics” Chaos10(4), 780—790.
30.
Lim, S.C. and Muniady, S.V., 2004, “Stochastic quantization of nonlocal fields,” Physics Letters A 324(5—6), 396—405.
31.
Magin, R.L., 2004, “Fractional calculus in bioengineering,” Critical Reviews in Biomedical Engineering32(1), 1—104.
32.
Magin, R.L., 2004, “Fractional calculus in bioengineering, Part 2” Critical Reviews in Biomedical Engineering32(2), 105—193.
33.
Magin, R.L., 2004, “Fractional calculus in bioengineering, Part 3,” Critical Reviews in Biomedical Engineering32(3/4), 194—377.
34.
Mainardi, F., 1996, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons and Fractals7(9), 1461—1477.
35.
Mainardi, F., Pagnini, G., and Gorenflo, R., 2003, “Mellin transform and subordination laws in fractional diffusion processes,” Fractional Calculus and Applied Analysis6(4), 441—459.
36.
Muslih, S. and Baleanu, D., 2005, “Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives,” Journal of Mathematical Analysis and Applications304(3), 599—606.
37.
Naber, M., 2004, “Time fractional Schrödinger equation” Journal of Mathemathical Physics45(8), 3339—3352.
38.
Nonnenmacher, T.F., 1990, “Fractional integral and differential equations for a class of Levy-type probability densities,” Journal of Physics A: Mathematical and General23(3), L697S—L700S.
39.
Rabei, E., Alhalholy, T., and Rousan, A., 2004, “Potential of arbitrary forces with Fractional derivatives,” International Journal of Theoretical Physics A 19(17—18), 3083—3092.
40.
Raspini, A., 2001, “Simple solutions of the fractional Dirac equation of order 3/2 ”, Physica Scripta64(1), 20—22.
Riewe, F., 1997, “Mechanics with fractional derivatives” Physical ReviewE55(3), 3581—3592.
43.
Scalas, E., Gorenflo, R., and Mainardi, F., 2004, “Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation,” Physics ReviewE69, 011107.
