The cycloid as brachistochrone: A one-page proof, from first principles, without calculus

TL;DR

This paper presents a straightforward, calculus-free proof of the brachistochrone problem using fundamental principles, and offers an optical interpretation of the solution.

Contribution

It provides a novel, minimalistic proof of the brachistochrone problem from first principles without calculus, and introduces an optical perspective.

Findings

Proof of the brachistochrone problem without calculus

Optical interpretation of the brachistochrone curve

Simplified understanding of the problem's principles

Abstract

Johann Bernoulli's optical solution of the brachistochrone problem is rebuilt on underlying (non-optical) principles. An "optical interpretation" is given afterwards.