An experimental numerics approach to the terrestrial brachistochrone

Abstract

We revisit the classical and solved problem of the terrestrial brachistochrone, the fastest path between two points in earth's gravitation field, by an approach we refer to as experimental numerics.
By this term we mean arriving at a qualified guess by deliberately taking inspiration from numerical results that are easily available.
Since in many cases verification is easier than derivation, this approach may have some educational merits.
Current software tools such as Jupyter Notebooks blend coding with documentation and allow leveraging this approach to enable new ways in modern teaching.
The intended audience are graduate students with prior knowledge of multivariate calculus, ordinary differential equations (ODEs), calculus of variations and classical physics, particularly mechanics.