The Brachistochrone

What just happened?

Three balls set off simultaneously from point $A$ and roll frictionlessly (gravity only) to point $B$, which lies lower and to the side. The ball on the cycloid arrives first — even though its path is the longest. The intuitive “straight line” loses.

The famous challenge of 1696:

The brachistochrone (Gr. brachistos — shortest, chronos — time) is one of the most famous problems in the history of mathematics, posed as a challenge by Johann Bernoulli. The question: along which path should a ball slide (without friction) to travel from $A$ to $B$ in the shortest possible time? Intuition suggests the shortest path — a straight line — but the answer is surprising.

The answer: the cycloid

The fastest path has the shape of a cycloid — the curve traced by a point on the rim of a rolling circle. Its parametric equations (rolling angle $\theta$, circle radius $a$):

Although the cycloid is longer than the straight line, its steep start lets the ball gain speed quickly — making the total time shorter.

Why it works — energy and time:

By conservation of energy (starting from rest), the speed depends only on the vertical drop $y$:

The total travel time is the integral over the path length — the sooner $v$ grows, the smaller its contribution to the time:

The cycloid drops almost vertically at the start, so $v$ grows immediately and the integral $T$ comes out smallest. In the simulation we computed this integral numerically for each path — the cycloid wins deterministically.

The birth of the calculus of variations:

The problem was solved independently by, among others, Newton, Leibniz, Jakob and Johann Bernoulli, and de l’Hôpital. It gave rise to the calculus of variations — a field that seeks not a number but an entire function (here: the shape of the path) that minimizes some quantity.

A curiosity — the tautochrone:

The cycloid is also a tautochrone (Gr. “same time”): a ball released from any point of such a curve reaches the bottom in the same time, no matter how high we place it.

Invariant:

The cycloid minimizes the travel TIME — not the path length. The shortest path (straight line) and the shortest time (cycloid) are two different curves.

Why this matters for your exam:

This is a textbook example of optimization: we ask not “how much” but “what shape” gives the smallest value. It combines functions, derivatives and integrals with physics (gravity, energy). Remember the mechanism: the speed $v=\sqrt{2gy}$ depends on the drop, while the time $T=\int ds/v$ rewards a fast, steep start.

Remember: the fastest descent from $A$ to $B$ is the cycloid, not the straight line — a longer path but a shorter time, because the steep start speeds the ball up sooner.