The Frictional Brachistochrone

TL;DR

This paper introduces a frictional brachistochrone model for granular materials that accounts for Coulomb frictional energy loss, revealing how increasing friction straightens the optimal path and suggesting industrial applications.

Contribution

It presents a novel frictional brachistochrone model that incorporates evolving path curvature and Coulomb friction, extending classical brachistochrone problems to granular materials.

Findings

Path becomes less curved with higher friction

Straight brachistochrone emerges at high friction

Potential industrial applications discussed

Abstract

Here, I construct an elegant frictional brachistochrone for a mass point motion of a granular material with the Coulomb frictional energy dissipation that inherently includes the evolving path curvature. The simple model reveals several striking mechanical phenomena. It is applicable to any frictional particle. With increasing friction, the particle path becomes less and less curved until a straight brachistochrone is attained in the limit of sufficiently high friction. The existence of the straight-brachistochrone is phenomenal. Some potential industrial applications of the frictional brachistochrone are considered.