Nonlinear Dynamics

, Volume 53, Issue 1–2, pp 67–74

# On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative

Original Paper

## Abstract

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.

## Keywords

Fractional Lagrangians Fractional calculus Fractional Riemann–Liouville derivative Fractional Euler–Lagrange equations Faà di Bruno formula

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## Authors and Affiliations

• Dumitru Baleanu
• 1
• Sami I. Muslih
• 2
• 3
• Eqab M. Rabei
• 4
• 5
1. 1.Department of Mathematics and Computer Sciences, Faculty of Arts and SciencesÇankaya UniversityAnkaraTurkey
2. 2.Department of PhysicsAl-Azhar UniversityGazaPalestine
3. 3.International Center for Theoretical PhysicsTriesteItaly
4. 4.Department of ScienceJerash Private UniversityJerashJordan
5. 5.Department of PhysicsMutah UniversityKarakJordan