# Necessary optimality conditions for the calculus of variations on time scales

## TL;DR

This paper extends the calculus of variations on time scales by establishing necessary optimality conditions for higher-order delta derivatives and problems with delta-differential constraints, broadening the theoretical framework.

## Contribution

It generalizes previous results by deriving necessary conditions for higher-order variational problems and those with delta-differential side conditions on time scales.

## Key findings

- Derived necessary optimality conditions for higher-order delta derivatives.
- Established conditions for variational problems with delta-differential constraints.
- Extended the calculus of variations framework on time scales.

## Abstract

We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems of the calculus of variations with delta-differential side conditions (Lagrange problem of the calculus of variations on time scales).