Brachistochrone of a Spherical Uniform Mass Distribution

TL;DR

This paper solves the brachistochrone problem for a particle moving through a uniform spherical mass, linking it to gravity elevator concepts and comparing it with the classical uniform gravity case.

Contribution

It provides a novel solution to the brachistochrone problem within a spherical mass distribution, extending classical results to more realistic gravitational environments.

Findings

Derived the brachistochrone curve for spherical mass distributions

Connected the solution to gravity elevator phenomena

Compared results with classical uniform gravity brachistochrone

Abstract

We solve the brachistochrone problem for a particle travelling through a spherical mass distribution of uniform density. We examine the connection between this problem and the popular "gravity elevator" result. The solution is compared to the well known brachistochrone problem of a particle in a uniform gravitational field.