Necessary Optimality Conditions for Fractional Action-Like Integrals of Variational Calculus with Riemann-Liouville Derivatives of Order $(\alpha,\beta)$

TL;DR

This paper derives Euler-Lagrange equations for fractional variational integrals involving Riemann-Liouville derivatives of orders (,eta), expanding the calculus of variations to fractional derivatives with potential applications.

Contribution

It introduces necessary optimality conditions for fractional action-like integrals with Riemann-Liouville derivatives of arbitrary positive orders (,eta), a recent development in fractional calculus.

Findings

Derived Euler-Lagrange equations for fractional derivatives

Discussed implications of fractional variational calculus

Extended calculus of variations to fractional orders

Abstract

We derive Euler-Lagrange type equations for fractional action-like integrals of the calculus of variations which depend on the Riemann-Liouville derivatives of order $(\alpha,\beta)$, $\alpha > 0$, $\beta > 0$, recently introduced by J. Cresson and S. Darses. Some interesting consequences are obtained and discussed.