Variational problems with fractional derivatives: Invariance conditions and Nother's theorem

A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal transformation group (basic Nother's identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of Nother's theorem leading to conservation laws for fractional Euler-Lagrangian equation is obtained as well. Results are illustrated by several concrete examples. Finally, an approximation of a fractional Euler-Lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws. (C) 2008 Elsevier Ltd. All rights reserved.