Short communication A new method of finding the fractional Euler–Lagrange and Hamilton equations within Caputo fractional derivatives Dumitru Baleanu a,b, * , Juan I. Trujillo c a Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey b Institute of Space Sciences, P.O. Box, MG-23, R 76900, Magurele-Bucharest, Romania c University of La Laguna, Departamento de Análisis Matemático, 38271 La Laguna, Tenerife, Spain article info Article history: Received 17 April 2009 Accepted 3 May 2009 Available online 13 May 2009 PACS: 11.10. Ef Keywords: Fractional Lagrangians Fractional calculus Fractional Caputo derivative Fractional Euler–Lagrange equations Faà di Bruno formula abstract In this paper, we have investigated the fractional Caputo derivative of a composition func- tion. The obtained results were applied to investigate the fractional Euler–Lagrange and Hamilton equations for constrained systems. The approach was applied within an illustrative. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The fractional calculus represents a generalization of ordinary differentiation and integration to arbitrary order [1–3].The fractional derivatives have the property that they are the infinitesimal generators of a class of translation invariant convolution semigroups appearing universally as attractors. The form of the fractional Leibniz rule and the fractional chain rule [4] as well as the fractional Taylor series [5,6] are different from the classical ones and their reflect the non-locality of the fractional operators. Despite of the above mentioned properties, the fractional calculus was used, as a powerful tool, for solving various problems involving the non-locality in many fields in both science and engineering (see for example, [7–16] and the references therein). During the last years, the fractional variational principles have been applied successfully in control theory [17–19] as well as in physics [20–29]. Particularly, for the constrained systems [30] the fractional Lagrangian and Hamiltonian formulations are still at the beginning of their development. By using the natural generalization of the classical variational principles, we obtain the fractional Euler–Lagrange equations which differ from the classical one except when the order of the fractional derivative is integer. An important issue was to obtain the fractional Euler–Lagrange equations for a given fractional Lagrangian and to construct the corresponding Hamiltonian. The non-locality of the fractional Lagrangian is embedded in the definition of the fractional derivatives. As a result, the fractional Euler–Lagrange equations involve both the left and the right derivatives and only in some specific cases we can solve the equations explicitly. For these reasons some other new techniques can be used to describe the fractional dynamics of a given system. Recently, the Hamiltonian formalism for non-local Lagrangians was investigated in [31] and a relation of non-local theories with Ostrogradski’s formalism [32] was analyzed in [33–35]. 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.05.023 * Corresponding author. Address: Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey. Fax: +90 312 2868962. E-mail addresses: dumitru@cankaya.edu.tr (D. Baleanu), JTrujill@ullmat.es (J.I. Trujillo). Commun Nonlinear Sci Numer Simulat 15 (2010) 1111–1115 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns