  Fractional mechanics describes both conservative and non-conservative
systems. The fractional variational principles gained importance in studying
the fractional mechanics and several versions are proposed. In classical
mechanics the equivalent Lagrangians play an important role because they admit
the same Euler-Lagrange equations. By adding a total time derivative of a
suitable function to a given classical Lagrangian or by multiplying with a
constant the Lagrangian we obtain the same equations of motion.
  In this study, the fractional discrete Lagrangians which differs by a
fractional derivative are analyzed within Riemann-Liouville fractional
derivatives.
  As a consequence of applying this procedure the classical results are
reobtained as a special case. The fractional generalization of $Fa\grave{a}$ di
Bruno formula is used in order to obtain the concrete expression of the
fractional Lagrangians which differs from a given fractional Lagrangian by
adding a fractional derivative. The fractional Euler-Lagrange and Hamilton
equations corresponding to the obtained fractional Lagrangians are investigated
and two examples are analyzed in details.
