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

TL;DR

This paper establishes that Nabla-differentiability of the forward jump operator Sigma is essential for deriving a differential Euler-Lagrange equation in calculus of variations on time scales, enabling a Noether-type theorem for conserved quantities.

Contribution

It proves the necessity of Nabla-differentiability of Sigma for differential Euler-Lagrange equations and introduces a Noether theorem with explicit constants of motion.

Findings

Nabla-differentiability of Sigma is a sharp condition for differential Euler-Lagrange equations.

Derived a Noether-type theorem providing explicit conserved quantities.

Clarified the role of the jump operator's differentiability in calculus of variations on time scales.

Abstract

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.