Nonshifted calculus of variations on time scales with ∇-differentiable σ

In calculus of variations on general time scales, an Euler–Lagrange equation of integral form is usually derived in order to characterize the critical points of nonshifted Lagrangian functionals, see e.g., Ferreira et al. (2011) [13]. In this paper, we prove that the ∇-differentiability of the forward jump operator σ is a sharp assumption on the time scale in order to ∇-differentiate this integral Euler–Lagrange equation. This procedure leads to an Euler–Lagrange equation of differential form. Furthermore, from this differential form, we prove a Noether-type theorem providing an explicit constant of motion for Euler–Lagrange equations admitting a symmetry.