A classical problem in the calculus of variations is to fi­nd the shape of a curve C  such that a bead, under the influence of gravity, will slide from point A(0, 0) to point B(x₁, y₁) in the least time. See the following figure. It can be shown that a nonlinear differential for the shape y(x) of the path is y[1 + (y')²] = k, where k is a constant. First solve for dx in terms of y and dy, and then use the substitution y = k sin²Î¸ to obtain a parametric form of the solution. The curve turns out to be a cycloid.

Transcribed Image Text:

A(0, 0) х bead B(x1, y1) mg y+