On Fractional Variational Problems which Admit Local Transformations

TL;DR

This paper extends Noether's second theorem to fractional variational problems involving Caputo derivatives, demonstrating invariance under local transformations with arbitrary functions, with applications to electromagnetic field Lagrangians.

Contribution

It introduces a fractional version of Noether's second theorem for problems with local symmetries involving Caputo derivatives.

Findings

Extended Noether's second theorem to fractional calculus

Applied the theorem to electromagnetic field Lagrangian

Showed invariance under local transformations with arbitrary functions

Abstract

We extend the second Noether theorem to fractional variational problems which are invariant under infinitesimal transformations that depend upon $r$ arbitrary functions and their fractional derivatives in the sense of Caputo. Our main result is illustrated using the fractional Lagrangian density of the electromagnetic field.