Brachistochrone

The classical problem asks which track connecting a starting point and a lower destination lets a small bead, released from rest, arrive in the shortest time. The idealized model assumes gravity is the only driving force and that the bead slides without friction. The quantity being minimized is travel time, not track length: a route that descends steeply at first can build speed early and outperform a shorter or more direct route.

Under the classical assumptions, the optimal track is a segment of a cycloid, the path traced by a point on the rim of a circle rolling along a straight line. In one common orientation, its parametric form is x = a(t − sin t), y = a(1 − cos t), where a sets the scale and t is a parameter. The segment is chosen to pass through the required endpoints. Depending on their relative positions, the path may initially dip steeply; a cycloidal arc can even contain a short rise while still minimizing total travel time.

For a particle starting from rest and dropping a vertical distance y in a uniform gravitational field, conservation of energy gives speed v = √(2gy), where g is gravitational acceleration. A curve's travel time is the integral of its small arc lengths divided by the speed at each point. Since a route that gains vertical drop early allows the particle to move faster over the remainder, optimizing elapsed time gives a different answer from minimizing distance. Variational calculus treats the unknown curve itself as the object to optimize; applying its methods yields the cycloid.

Galileo discussed related quickest-descent questions in the 17th century, but his suggested circular path was not the solution to the full problem. Johann Bernoulli issued the well-known challenge in 1696. Historical accounts report solutions by Johann and Jakob Bernoulli, Gottfried Wilhelm Leibniz, Isaac Newton, and Guillaume de l'Hôpital; Newton's solution was published anonymously in 1697. The challenge became a landmark in the emergence of the calculus of variations. The result is model-dependent: friction, rolling rather than sliding, air resistance, or nonuniform gravity changes the physical problem and may change its optimal path.

The cycloid is also associated with the tautochrone problem: in the idealized setting, the time for a particle to descend to the lowest point of an inverted cycloidal track is independent of its release position on the arc. This is a separate property from the brachistochrone's least-time path between two specified endpoints, although both problems lead to the same curve family.