Variational Calculus with Conformable Fractional Derivatives

TL;DR

This paper develops invariant conditions and Noether's theorem for conformable fractional calculus of variations, simplifying the formulation of physical action principles involving frictional forces using conformable derivatives.

Contribution

It introduces new invariance conditions and fractional Noether's theorem for conformable derivatives, and applies these to formulate a simpler action principle for particles with friction.

Findings

Derived fractional Noether's theorem for conformable derivatives

Established invariant conditions for fractional optimal control

Presented a simplified action principle for particles under friction

Abstract

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.