Fractional Powers of Derivatives in Classical Mechanics

TL;DR

This paper introduces a fractional calculus approach to classical mechanics, generalizing equations of motion through fractional derivatives to model dissipative processes, and provides exact solutions for simple systems.

Contribution

It presents a novel fractional generalization of classical equations of motion using fractional powers of derivative operators, extending the modeling capabilities for dissipative phenomena.

Findings

Exact solutions for simple classical systems with fractional derivatives

Fractional equations generalize classical models to include dissipation

Fractional derivatives as fractional powers of operators provide new insights

Abstract

Fractional analysis is applied to describe classical dynamical systems. Fractional derivative can be defined as a fractional power of derivative. The infinitesimal generators {H, .} and L=G(q,p) \partial_q+F(q,p) \partial_p, which are used in equations of motion, are derivative operators. We consider fractional derivatives on a set of classical observables as fractional powers of derivative operators. As a result, we obtain a fractional generalization of the equation of motion. This fractional equation is exactly solved for the simple classical systems. The suggested fractional equations generalize a notion of classical systems to describe dissipative processes.