A unified approach to the calculus of variations on time scales

TL;DR

This paper introduces a comprehensive framework for the calculus of variations on time scales, unifying delta and nabla approaches and deriving necessary and sufficient optimality conditions with illustrative examples.

Contribution

It presents a generalized method that unifies delta and nabla calculus of variations on time scales, including new optimality conditions and examples.

Findings

Unified Euler-Lagrange conditions derived

Applicable to both delta and nabla integrals

Illustrated with simple, concrete examples

Abstract

In this work we propose a new and more general approach to the calculus of variations on time scales that allows to obtain, as particular cases, both delta and nabla results. More precisely, we pose the problem of minimizing or maximizing the composition of delta and nabla integrals with Lagrangians that involve directional derivatives. Unified Euler-Lagrange necessary optimality conditions, as well as sufficient conditions under appropriate convexity assumptions, are proved. We illustrate presented results with simple examples.