The brachistochrone

Four beads are let go together at A and slide without friction to B, each on its own track: the ivory straight line, the ice-blue arc of a circle with B at its lowest point, a coral fourth track and the gold cycloid. The corner gives the time measured along each track: the gold cycloid is quickest, and the straight line, though shortest, is slower. Drag B somewhere else and race again; the quickest is always the cycloid through A and B. When B is far off and only a little lower, the cycloid plunges steeply, dips below B and climbs back up; when B is nearly straight below A, the cycloid is almost straight too. The fourth track pushes the depth at every point from the straight line towards the cycloid: s = 0 is the line, s = 1 is the cycloid, and a larger s dives deeper. The small graph at the right plots its time against s; the lowest point is exactly at s = 1, and the graph is flat there: nudge s a little and the time barely changes. Diving deeper than the cycloid only makes it slower. The fine lines spreading from A are the synchrones, joining the places a bead let go at A can reach in the same time by the quickest route; the gold bead stays on the leading one all the way to B.