The Variational Calculus on Time Scales

TL;DR

This paper reviews a general approach to the calculus of variations on time scales, unifying discrete, quantum, and continuous cases, and encompassing both delta and nabla integral minimizations.

Contribution

It introduces a comprehensive framework that generalizes existing methods, allowing derivation of both delta and nabla calculus of variations results as special cases.

Findings

Unified approach covers delta and nabla calculus of variations

Enables derivation of known results as particular cases

Provides a more general theoretical foundation

Abstract

The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we review a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.