  In calculus of variations on general time scales, an integral Euler-Lagrange
equation is usually derived in order to characterize the critical points of non
shifted Lagrangian functionals, see e.g. [R.A.C. Ferreira and co-authors,
Optimality conditions for the calculus of variations with higher-order delta
derivatives, Appl. Math. Lett., 2011].
  In this paper, we prove that the Nabla-differentiability of the forward jump
operator Sigma is a sharp assumption in order to obtain an Euler-Lagrange
equation of differential form. Furthermore, this differential form allows us to
prove a Noether-type theorem providing an explicit constant of motion for
differential Euler-Lagrange equations admitting a symmetry.
