ve such that a bead, under the influence of gravity, will

Chapter , Problem 10RP

(choose chapter or problem)

A classical problem in the calculus of variations is to find the shape of a curve \(\mathscr{C}\) such that a bead, under the influence of gravity, will slide from point A(0, 0) to point \(B\left(x_{1}, y_{1}\right)\) in the least time. See Figure 3.R.2. It can be shown that a nonlinear differential for the shape y(x) of the path is \(y\left[1+\left(y^{\prime}\right)^{2}\right]=k\), where k is a constant. First solve for dx in terms of y and dy, and then use the substitution \(y=k \sin ^{2} \theta\) to obtain a parametric form of the solution. The curve  turns out to be a cycloid.

Text Transcription:

mathscr{C}

B(x_1, y_1)

y[1  (y^prime)^2 ] = k

y = k sin^2 u