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

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.

This is a preprint of a paper whose final and definite form is published in Journal of Mathematical Analysis and Applications

References in corpus (6)

• Noether's Theorem on Time Scales
• Calculus of Variations on Time Scales with Nabla Derivatives
• Higher-Order Calculus of Variations on Time Scales
• Noether's symmetry theorem for nabla problems of the calculus of variations
• Optimality conditions for the calculus of variations with higher-order delta derivatives
• A Study of Diamond-alpha dynamic equations on regular time scales