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Non shifted calculus of variations on time scales with Nabla-differentiable Sigma

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.