A detailed analysis of the brachistochrone problem

TL;DR

This paper provides a comprehensive and detailed analysis of the classical brachistochrone problem, exploring the mathematical optimization involved in finding the quickest descent path under gravity.

Contribution

It offers an in-depth, step-by-step examination of the brachistochrone problem, filling gaps in detailed derivations and explanations not previously available.

Findings

Derivation of the optimal trajectory using calculus of variations

Explicit solution to the minimization problem

Clarification of the mathematical properties of the solution

Abstract

If A and B are two points in the plane, with B lower and to the right of A, then we may consider the trajectory of an object travelling from A to B under the influence of gravity. The search for the trajectory minimising the time taken by the object gives rise to a mathematical optimisation problem involving an indefinite integral. Although the solution of this problem is known, a full detailed handling of the problem does not seem to be available. The aim of this article is to provide such a detailed study.