Brachistochrone

The brachistochrone problem

In June 1696, Johann Bernoulli published a challenge in the Acta Eruditorum: given two points A and B, with A higher than B, find the curve along which a body sliding under gravity alone (no friction) descends from A to B in the shortest possible time. He called it the brachistochrone, from the Greek brachistos (shortest) and chronos (time). The straight line between the two points is the shortest distance, but it is not the fastest path. A curve that dips steeply at first, gaining speed early, and then flattens out toward the end, can beat the straight line even though the total distance is longer. The answer is a cycloid — the curve traced by a point on the rim of a rolling wheel.

Why the cycloid wins

The key insight is that speed depends on how far the bead has fallen: by conservation of energy, v = √(2gy), where y is the vertical distance below the starting point. A path that drops steeply at the beginning converts potential energy into kinetic energy early, so the bead is traveling fast for most of the journey. The straight line drops gradually and uniformly, so the bead is slow at the start when it has the most horizontal distance to cover. The cycloid optimizes this tradeoff: it falls steeply enough to gain speed quickly but not so steeply that it wastes distance going straight down. It is the unique solution to the variational problem of minimizing descent time.

The tautochrone property

The cycloid has a second remarkable property, discovered by Christiaan Huygens in 1659, decades before Bernoulli’s challenge. A bead released from any point on an inverted cycloid and sliding to the bottom under gravity will always arrive in the same amount of time, regardless of where it started. This is the tautochrone property (from tauto, same, and chronos, time). It is not obvious that the curve of fastest descent and the curve of equal descent time should be the same curve. That they are is one of the most beautiful coincidences in classical mechanics. Huygens used this property to design a pendulum clock with cycloidal cheeks that would keep perfect time regardless of the amplitude of the swing.

Bernoulli’s challenge and Newton’s response

Bernoulli’s challenge was deliberately aimed at the leading mathematicians of Europe, and especially at Isaac Newton. The story goes that Newton received the problem at 4 PM after a long day at the Royal Mint, solved it by midnight, and sent his solution anonymously. Bernoulli reportedly recognized it, saying “I recognize the lion by his claw.” Solutions were also submitted by Gottfried Leibniz, Jakob Bernoulli (Johann’s brother), and the Marquis de l’Hôpital. The problem became a founding moment for the calculus of variations — the branch of mathematics concerned with finding functions that optimize functionals (integrals that depend on the shape of a curve).

The mathematics

The cycloid is defined parametrically. For a wheel of radius r rolling along a horizontal surface, a point on the rim traces the curve:

x = r(θ − sinθ)
y = r(1 − cosθ)

where θ ranges from 0 to 2π for one complete arch. The radius r is chosen so the cycloid passes through the desired endpoint. The time of descent along the full cycloid arch is T = π√(r/g), independent of the horizontal or vertical distance (as long as the endpoint lies on the cycloid).

Connection to the Euler-Lagrange equation

The brachistochrone is solved by minimizing the integral T = ∫ ds / v = ∫ √(1 + y′²) / √(2gy) dx. The integrand does not depend on x explicitly, so by the Beltrami identity (a special case of the Euler-Lagrange equation), the first integral gives a conserved quantity that reduces the problem to a first-order ODE. The solution to this ODE is the cycloid. This approach — expressing a physical problem as the extremization of an integral, deriving the differential equation that the extremizing function must satisfy, and solving it — is the method of the calculus of variations, and the brachistochrone is the problem that gave birth to it.