Variational Problems with Fractional Derivatives: Euler-Lagrange   Equations
  We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense. 
Teodor M. Atanackovic, Sanja Konjik, Stevan Pilipovic
math.FA