The brachistochrone problem: find the curve of fastest descent between two points under gravity. Johann Bernoulli's 1696 challenge; Newton solved it overnight. The answer is a cycloid — the path traced by a point on a rolling circle. The Euler-Lagrange equation from variational calculus δ∫L dt = 0 yields this exact solution. Drag the endpoint to compare.
Cycloid: — s | Straight: — s | Parabola: — s
The cycloid always wins, no matter the endpoint. This is Bernoulli's principle of least time — equivalent to Fermat's principle in optics.