Chapter 2.3, Problem 34P

Brachistochrone Problem. One of the famous problems in the history of mathematics is the brachistochronproblem: to find the curve along which a particle will slidewithout friction in the minimum time from one given point $P$ to another $Q$, the second point being lower than the firstbut not directly beneath it (see Figure $\mathrm{2.3.6})$. This problemwas posed by Johann Bernoulli in $1696$ as a challenge problem to the mathematicians of his day. Correct solutions werefound by Johann Bernoulli and his brother Jakob Bernoulliand by Isaac Newton, Gottfried Leibniz, and the Marquis deL’Hopital. The brachistochrone problem is important in the development of mathematics as one of the forerunners of thecalculus of variations.

In solving this problem, it is convenient to take the originas the upper point $P$ and to orient the axes as shown in (Figure $\mathrm{2.3.6})$ The lower point $Q$ has coordinates $\left(x_0,y_0\right)$ It is then

possible to show that the curve of minimum time is given by a function $y=\varphi \left(x\right)$ that satisfies the differential equation

$\left(1+y{\text{'}}^2\right)y=k^2,$ …. (i)

Where $k^2$ is a certain positive constant to be determined later.

Solve Eq. (i) for $y\text{'}$. Why is it necessary to choose the positive square root?

Introduce the new variable $t$ by the relation

$y=k^2{\sin }^{2}t$…. (ii)

Show that equation found in part (a) then takes the form

$2k^2{\sin }^{2}tdt=dx$…. (iii)

Letting $\theta =2t,$ then show that the solution of Eq. (iii) for which $x=0$ when $y=0$ is given by

$x=k^2\left(\theta -\sin \theta \right)/2,$

…. (iv)

$x=k^2\left(1-\cos \theta \right)/2.$

Equations (iv) are parametric equations of the solution of Eq. (i) that passes through $\left(0,0\right)$. The graph of Eqs. (iv) is called a cycloid.

If we make a proper choice of the constant $k,$ then the cycloid also passes through the point $\left(x_0,y_0\right)$ and is the solution of the brachistochrone problem. Find $k$ if $x_0=1$ and $y_0=2$.