Calculus of Variations on Time Scales with Nabla Derivatives

TL;DR

This paper establishes a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving higher-order nabla derivatives, broadening the theoretical framework of calculus of variations.

Contribution

It introduces a new fundamental lemma of the calculus of variations on time scales and extends the Euler-Lagrange condition to higher-order nabla derivatives.

Findings

Derived a necessary optimality condition for higher-order nabla derivatives.

Developed a more general fundamental lemma for calculus of variations on time scales.

Extended the theoretical framework for variational problems on time scales.

Abstract

We prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of variations on time scales.