Bernoulli's brachistochrone problem

Statement

Find the curve of fastest descent between two points under gravity (frictionless). Among all curves joining two points, the brachistochrone minimizes the time for a particle to travel from one point to the other under gravity alone.

Formula

The cycloid is the curve of fastest descent.

Equation

x = r(θ − sin θ), y = r(1 − cos θ) (cycloid)

Proof sketch

Johann Bernoulli posed the problem in Acta Eruditorum (1696). Newton, Leibniz, Jacob Bernoulli, and Johann Bernoulli himself all solved it independently. The solution is a cycloid—the curve traced by a point on a rolling circle. The problem led to the development of the calculus of variations.