A classical problem in the calculus of variations is to find the shape of a curve such...

A classical problem in the calculus of variations is to find the shape of a curve such that a bead, under the influence of gravity, will slide from in the least time. See Figure 3.24. It can be shown that the differential equation for the shape of the path is where k is a constant. First solve for dx in terms of y and dy, and then use the substitution to obtain the parametric form of the solution. The curve turns out to be a cycloid.