Applications of Fractional Calculus to Newtonian Mechanics

TL;DR

This paper explores how fractional calculus can extend Newtonian mechanics, offering new models for force laws and gravitational potentials that could help explain galactic rotation curves and dark matter.

Contribution

It introduces fractional calculus into Newtonian mechanics, generalizing force laws and gravitational potentials with potential applications in astrophysics.

Findings

Generalized Newton's second law using fractional derivatives

Derived fractional gravitational potential and orbital velocities

Potential to model galactic rotation curves and dark matter effects

Abstract

We investigate some basic applications of Fractional Calculus (FC) to Newtonian mechanics. After a brief review of FC, we consider a possible generalization of Newton's second law of motion and apply it to the case of a body subject to a constant force. In our second application of FC to Newtonian gravity, we consider a generalized fractional gravitational potential and derive the related circular orbital velocities. This analysis might be used as a tool to model galactic rotation curves, in view of the dark matter problem. Both applications have a pedagogical value in connecting fractional calculus to standard mechanics and can be used as a starting point for a more advanced treatment of fractional mechanics.