Nonlinear Dyn (2009) 58: 385–391 DOI 10.1007/s11071-009-9486-z ORIGINAL PAPER Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations Mohamed A.E. Herzallah · Dumitru Baleanu Received: 1 October 2008 / Accepted: 26 February 2009 / Published online: 18 March 2009 © Springer Science+Business Media B.V. 2009 Abstract This paper presents the fractional order Euler–Lagrange equations and the transversality con- ditions for fractional variational problems with frac- tional integral and fractional derivatives defined in the sense of Caputo and Riemann–Liouville. A fractional Hamiltonian formulation was developed and some il- lustrative examples were treated in detail. Keywords Fractional integral · Fractional derivative · Fractional calculus of variations · Fractional Hamiltonian equations 1 Introduction Fractional calculus is one of the generalizations of the classical calculus and it has been used successfully in Dumitru Baleanu on leave of absence from Institute of Space Sciences, P.O. Box MG-23, R 76900, Magurele-Bucharest, Romania. M.A.E. Herzallah Faculty of Science, Zagazig University, Zagazig, Egypt e-mail: m_herzallah75@hotmail.com D. Baleanu ( ) Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey e-mail: dumitru@cankaya.edu.tr D. Baleanu e-mail: baleanu@venus.nipne.ro various fields of science and engineering [12, 13, 18, 19, 23, 27, 30]. Several fields of application of fractional differen- tiation and fractional integration are already well es- tablished, some others have just begun. Many appli- cations of fractional calculus can be found in turbu- lence and fluid dynamics, stochastic dynamical sys- tem, plasma physics and controlled thermonuclear fu- sion, nonlinear control theory, image processing, non- linear biological systems, astrophysics, etc. [1, 5, 8, 11, 12, 15–18, 20, 22, 23, 26]. One of the areas that recently emerged within frac- tional framework and which is being subject to in- tensive research is the calculus of variations and re- spectively the fractional Euler–Lagrange type equa- tions. Real integer variational calculus plays a signif- icant role in many areas of science, engineering and applied mathematics [28, 29]. In many applications, it is used to obtain the laws governing the physics of systems and boundary/terminal conditions. It has been the starting point for various numerical schemes such as the Riesz, finite difference and finite element meth- ods. In optimal control, it is used to obtain the differ- ential equations and the terminal conditions for opti- mal trajectory of a system (see [3] and the references therein). The fractional calculus of variations was introduced by Riewe in [24, 25] where he developed the non- conservation Lagrangian, Hamiltonian, and other con- cepts of classical mechanics using fractional calcu- lus. Klimek gave another approach by considering