Sliding with Friction and The Brachistochrone Problem

TL;DR

This paper investigates the brachistochrone problem considering Coulomb friction, deriving optimal curves that minimize descent time and generalize the cycloid, supported by numerical calculations.

Contribution

It introduces a parametric equation for optimal curves under Coulomb friction, extending the classical brachistochrone problem.

Findings

Derived a generalized brachistochrone curve with Coulomb friction

Numerical calculations confirm the curve minimizes descent time

The curve reduces to the cycloid when friction is absent

Abstract

We analyze the motion of a particle in the gravity field along a family of differentiable curves taking into account the Coulomb friction forces. A parametric equation of the optimal curves is given that generalizes the cycloid one in this case. The results of numerical calculations in the Mathcad program show that the found curve minimize the descent time for a given friction coefficient and can claim to be a brachistochrone with Coulomb friction.