  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 N\"{o}ther's identity) are obtained. These
conditions extend the classical results, valid for integer order derivatives. A
generalization of N\"{o}ther'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.
