Chapter 3, Problem 12RE

A classical problem in the calculus of variations is to find the shape of a curve $\mathcal{C}$ 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 Figure 3.R.3. 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 sin² θ to obtain a parametric form of the solution. The curve $\mathcal{C}$ turns out to be a cycloid.

FIGURE 3.R.3 Sliding bead in Problem 12