On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative
- 279 Downloads
Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics, the equivalent Lagrangians play an important role because they admit the same Euler–Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant, the Lagrangian we obtain are the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann–Liouville fractional derivatives. As a consequence of applying this procedure, the classical results are reobtained as a special case.
The fractional generalization of Faà di Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler–Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated, and two examples are analyzed in detail.
KeywordsFractional Lagrangians Fractional calculus Fractional Riemann–Liouville derivative Fractional Euler–Lagrange equations Faà di Bruno formula
Unable to display preview. Download preview PDF.
- 10.Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997) Google Scholar
- 19.Magin, R.L.: Fractional calculus in bioengineering, part 3. Critic. Rev. Biomed. Eng. 32(3/4), 194–377 (2004) Google Scholar
- 25.Cresson, J.: Fractional embedding of differential operators and Lagrangian systems. Preprint (2006) Google Scholar
- 31.Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004) Google Scholar
- 32.Baleanu, D.: Constrained systems and Riemann–Liouville fractional derivative. In: Proceedings of 1st IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, July 19–21, pp. 597–602 (2004) Google Scholar
- 33.Baleanu, D.: About fractional calculus of singular Lagrangians. In: Proceedings of IEEE International Conference on Computational Cybernetics ICCC, Wien, Austria, August 30–September 1, pp. 379–385 (2004) Google Scholar
- 34.Baleanu, D., Muslih, S.: New trends in fractional quantization method, 1st edn. Intelligent Systems at the Service of Mankind, vol. 2. U-Books, Augsburg (2005) Google Scholar
- 35.Baleanu, D.: Constrained systems and Riemann–Liouville fractional derivatives. In: Le Mehaute, A., Tenreiro Machado, J.A., Trigeassou, J.C., Sabatier, J. (eds.) Fractional Differentiation and Its Applications, pp. 69–80. U-Books, Augsburg (2005) Google Scholar
- 37.Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1993) Google Scholar
- 44.Bering, K.: A note of non-locality and Ostrogradski’s construction. Preprint hep-th/0007192 Google Scholar
- 47.Negri, L.J., da Silva Joseph, E.G.: s-equivalent Lagrangians in generalized mechanics. Phys. Rev. D 27, 2227–2232 (1986) Google Scholar
- 48.Santilli, M.: Foundations of Theoretical Mechanics I, II. Springer, New York (1977) Google Scholar