Spatial Rotation of the Fractional Derivative in Two Dimensional Space

TL;DR

This paper derives how fractional derivatives transform under spatial rotation in 2D, linking their physical interpretation across different coordinate systems and aiding the construction of invariant interaction terms.

Contribution

It provides the first explicit derivation of the rotational transformation properties of Riemann-Liouville and Caputo fractional derivatives in two-dimensional space.

Findings

Transformation properties for fractional derivatives under rotation are established.

Insights into the physical interpretation of fractional derivatives are discussed.

Potential for constructing observer-invariant interaction terms is highlighted.

Abstract

The transformation of the partial fractional derivatives under spatial rotation in $R^2$ are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed through fractional derivatives, with respect to different coordinate systems (observers). It is the hope that such understanding could shed light on the physical interpretation of fractional derivatives. Also it is necessary to able to construct interaction terms that are invariant with respect to equivalent observers.