The cycloid as brachistochrone: A one-page proof, from first principles, without calculus
Gavin R. Putland
TL;DR
The work reconstructs Bernoulli's brachistochrone without optical premises by modeling a moving cycloidal cylinder with horizontal speed $u$ and normal speed $v$, and deriving $v/u = \cos \phi$ along with $y = -2a \cos^2 \phi$ to obtain $\frac{1}{2}v^2 + \frac{u^2}{4a}y = 0$. Energy conservation under gravity yields $\frac{1}{2}v^2 + gy = 0$, and choosing $u = \sqrt{4ga}$ makes the two expressions for $v(y)$ coincide, identifying the brachistochrone as the inverted cycloid OPB that passes through the endpoints. The analysis connects to an optical interpretation by viewing the moving front as a wavefront in an isotropic but non-homogeneous medium, with Snell-like behavior for stratified media, but the core result remains a geometric construction of the least-time path. The paper thus provides a self-contained, first-principles proof that the cycloid is the curve of fastest descent in a uniform gravitational field, with an explicit link to the classical optical solution and a broader interpretation in wavefront language.
Abstract
Johann Bernoulli's optical solution of the brachistochrone problem is rebuilt on underlying (non-optical) principles. An "optical interpretation" is given afterwards.