The contingent epiderivative and the calculus of variations on time scales

TL;DR

This paper introduces the contingent epiderivative as a new differentiation tool to unify delta and nabla approaches in the calculus of variations on time scales, leading to generalized Euler-Lagrange conditions.

Contribution

It proposes the contingent epiderivative for the calculus of variations on time scales, unifying existing delta and nabla methods and deriving generalized optimality conditions.

Findings

Unified delta and nabla approaches using the contingent epiderivative

Derived generalized Euler-Lagrange conditions for variational problems

Reproduced recent delta and nabla results as special cases

Abstract

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists in applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler-Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results.