Brachistochrone

This simulator lets you numerically investigate the classical brachistochrone problem: determining the path along which a particle, subject only to uniform gravity and starting from rest, takes the least time to travel from point A to point B.

The least-time path is a cycloid, described by the parametric equations $$x(\theta)=R(\theta-\sin\theta)$$ and $$y(\theta)=R(1-\cos\theta),$$ where $R$ is the generating radius and $\theta$ describes the motion along the curve.

Drag point B and the green control points 1 and 2 to change the path. Enable “Brachistochrone” to display the theoretical least-time cycloid, and use the examples to compare the straight line, the approximate brachistochrone, and the approximate tautochrone.

The curve is the solution to a variational problem: among all possible paths connecting the two points, the cycloid minimizes travel time—even outperforming the straight line, which minimizes only distance. The simulation compares the descent time along the cycloid with that of other candidate curves.

Marco P. M. de Souza