The brachistochrone problem

In June 1696 Johann Bernoulli issued a challenge in the journal Acta Eruditorum to «the sharpest mathematicians in the world»: given two points $A$ and $B$ at different heights, find the curve along which a bead, starting from rest and sliding without friction under gravity, reaches $B$ in the shortest time. This is the brachistochrone problem (from the Greek bráchistos, «shortest», and chrónos, «time»).

The answer is surprising: it is not the straight segment, which is indeed the shortest path, but an arc of a cycloid, the curve traced by a point on the rim of a rolling wheel. On a steeper initial stretch the bead speeds up sooner, and this gain in velocity outweighs the greater length.

The greatest minds of the age responded: Johann and his brother Jakob Bernoulli, Gottfried Leibniz, the Marquis de l’Hôpital and, anonymously, Isaac Newton — who, the story goes, solved the problem in a single night; reading the unsigned solution, Johann Bernoulli is said to have exclaimed that he «recognised the lion by his claw».

The problem asked one to minimise not a number but an integral, $T = \int_A^B \sqrt{\frac{1+y'^2}{2gy}}\,dx,$ that is, a quantity depending on the whole unknown function $y(x)$. Its solution gave the decisive push to the birth of the calculus of variations, later systematised by Leonhard Euler and Joseph-Louis Lagrange: their Euler–Lagrange equation, which characterises the optimal curve, is itself a differential equation. Thus a riddle about the motion of a bead turned out to be one of the first great problems solved precisely with differential equations (Boyer; Dunham).

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Topics: Differential equations
Concepts: Differential equation
People: Bernoulli · Leibniz · L’Hôpital · Newton · Euler · Lagrange