The delta-nabla calculus of variations

TL;DR

This paper introduces a unified calculus of variations on time scales that encompasses both delta and nabla approaches, providing a more general framework for discrete, quantum, and continuous cases.

Contribution

It proposes a comprehensive approach to the calculus of variations on time scales, unifying delta and nabla methods into a single framework.

Findings

Unified framework for delta and nabla calculus of variations.

Applicable to discrete, quantum, and continuous cases.

Simplifies derivation of variational results across different time scales.

Abstract

The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we propose a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.