A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives. (English) Zbl 1221.34008

Baleanu, Dumitru; Trujillo, Juan I.

Commun. Nonlinear Sci. Numer. Simul. 15, No. 5, 1111-1115 (2010).

Summary: We investigate the fractional Caputo derivative of a composition function. The obtained results are applied to investigate the fractional Euler–Lagrange and Hamilton equations for constrained systems. The approach is applied within an illustrative.

MSC:

34A08	Fractional ordinary differential equations
26A33	Fractional derivatives and integrals
45J05	Integro-ordinary differential equations
70H03	Lagrange’s equations

Keywords:

fractional Lagrangians; fractional calculus; fractional Caputo derivative; fractional Euler; Lagrange equations; Faà di Bruno formula

References:

[1]	Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives – theory and applications (1993), Gordon and Breach: Gordon and Breach Linghorne, PA · Zbl 0818.26003
[2]	Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[3]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations (2006), Elsevier · Zbl 1092.45003
[4]	Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions (1979), Nauka: Nauka Moscow · Zbl 0515.33001
[5]	Hardy, G. H., Riemann’s form of Taylor’s series, J London Math Soc, 20, 48-57 (1954) · Zbl 0063.01925
[6]	Trujillo, J. J., On a Riemann-Liouville generalized Taylor’s formula, J Math Anal Appl, 231, 255-265 (1999) · Zbl 0931.26004
[7]	Magin, R. L., Fractional calculus in bioengineering (2006), Begell House Publisher, Inc.: Begell House Publisher, Inc. CT
[8]	Tenreiro Machado, J. A., A probabilistic interpretation of the fractional-order differentiation, Frac Calc Appl Anal, 8, 73-80 (2003) · Zbl 1035.26010
[9]	Lorenzo, C. F.; Hartley, T. T., Fractional trigonometry and the spiral functions, Nonlinear Dynam, 38, 23-60 (2004) · Zbl 1094.26004
[10]	Silva, M. F.; Tenreiro Machado, J. A.; Lopes, A. M., Modelling and simulation of artificial locomotion systems, Robotica, 23, 595-606 (2005)
[11]	Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional orders, fractals and fractional calculus in continoum mechanics (1997), Springer Verlag: Springer Verlag Wien and New York · Zbl 1438.26010
[12]	Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7, 9, 1461-1477 (1996) · Zbl 1080.26505
[13]	Mainardi, F.; Pagnini, G.; Gorenflo, R., Mellin transform and subordination laws in fractional diffusion processes, Frac Calc Appl Anal, 6, 4, 441-459 (2003) · Zbl 1083.60032
[14]	Raspini, A., Simple solutions of the fractional Dirac equation of order \(\frac{2}{3} \), Phys Scripta, 64, 20-22 (2001) · Zbl 1061.81518
[15]	Naber, M., Time fractional Schrödinger equation, J Math Phys, 45, 8, 3339-3352 (2004) · Zbl 1071.81035
[16]	Rabei, E.; Alhalholy, T.; Rousan, A., Potential of arbitrary forces with fractional derivatives, Int J Theor Phys A, 19, 17-18, 3083-3096 (2004) · Zbl 1080.70516
[17]	Agrawal, O. P., Formulation of Euler-Lagrange equations for fractional variational problems, J Math Anal Appl, 272, 368-379 (2002) · Zbl 1070.49013
[18]	Agrawal, O. P.; Baleanu, D., A Hamiltonian formulation and a direct numerical scheme for Fractional Optimal Control Problems, J Vibr Contr, 13, 9-10, 1269-1281 (2007) · Zbl 1182.70047
[19]	Tenreiro Machado, J. A., Discrete-time fractional-order controllers, Frac Calc Appl Anal, 4, 47-66 (2001) · Zbl 1111.93307
[20]	Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys Rev E, 53, 1890-1899 (1996)
[21]	Riewe, F., Mechanics with fractional derivatives, Phys Rev E, 55, 3581-3592 (1997)
[22]	Klimek, M., Lagrangian and Hamiltonian fractional sequential mechanics, Czech J Phys, 52, 1247-1253 (2002) · Zbl 1064.70013
[23]	Baleanu, D.; Muslih, S., Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys Scripta, 72, 2-3, 119-121 (2005) · Zbl 1122.70360
[24]	Muslih, S.; Baleanu, D., Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives, J Math Anal Appl, 304, 3, 599-606 (2005) · Zbl 1149.70320
[25]	Rabei, E. M.; Nawafleh, K. I.; Hijjawi, R. S.; Muslih, S. I.; Baleanu, D., The Hamilton formalism with fractional variational derivatives, J Math Anal Appl, 327, 2, 891-897 (2007) · Zbl 1104.70012
[26]	Baleanu, D.; Muslih, S. I., Formulation of Hamiltonian equations for fractional variational problems, Czech J Phys, 55, 6, 633-642 (2005) · Zbl 1181.70017
[27]	Muslih, S.; Baleanu, D.; Rabei, E., Hamiltonian formulation of classical fields within Riemann-Liouville, Phys Scripta, 73, 6, 436-438 (2006) · Zbl 1165.70310
[28]	Baleanu, D.; Avkar, T., Lagrangians with linear velocities within Riemann-Liouville fractional derivatives, Nuovo Ciment B, 119, 73-79 (2004)
[29]	Baleanu, D.; Muslih, S. I., About fractional supersymmetric quantum mechanics, Czech J Phys, 55, 9, 1063-1066 (2005) · Zbl 1465.81042
[30]	Henneaux, M.; Teitelboim, C., Quantization of gauge systems (1993), Princeton University · Zbl 0638.58041
[31]	Llosa, J.; Vives, J., Hamiltonian formalism for nonlocal Lagrangians, J Math Phys, 35, 2856-2877 (1994) · Zbl 0807.70014
[32]	Nesterenko, V. V., Singular Lagrangians with higher derivatives, J Phys A: Math Gen, 22, 1673-1687 (1989) · Zbl 0695.58014
[34]	Baleanu, D.; Trujillo, J. J., On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dynam, 52, 4, 331-335 (2008) · Zbl 1170.70328
[35]	Gomis, J.; Kamimura, K.; Llosa, J., Hamiltonian formalism for space-time noncummutative theories, Phys Rev D, 63, 945003-1-945003-6 (2001)