Chapter 1.4, Problem 68P

Figure 1.4.11 shows a bead sliding down a frictionless wire from point P to point Q. The brachistochrone problem asks what shape the wire should be in order to minimize the bead’s time of descent from P to Q. In June of 1696. John Bernoulli proposed this problem as a public challenge, with a 6-month deadline (later extended to Easter 1697 at George Leibniz’s request). Isaac Newton, then retired from academic life and serving as Warden of the Mint in London, received Bernoulli’s challenge on January 29, 1697. The very next day he communicated his own solution—the curve of minimal descent time is an arc of an inverted cycloid—to the Royal Society of London. For a modern derivation of this result, suppose the bead starts from rest at the origin P and let $y=y\left(x\right)$ be the equation of the desired curve in a coordinate system with the y -axis pointing downward. Then a mechanical analogue of Snell’s law in optics implies that $\frac{\sin \alpha }{v}=\text{constant},$ ......(i)

where $\alpha$ denotes the angle of deflection (from the vertical) of the tangent line to the curve—so $\alpha =y^{\prime }\left(x\right)$ (why?) and $v=\sqrt{2gy}$ is the bead’s velocity when it has descended a distance y vertically (from $KE=\frac{1}{2}m{v}^2=mgy=-\text{PE}$ ).

(a) First derive from Eq. (1) the differential equation $\frac{dy}{dx}=\sqrt{\frac{2a-y}{y}}$ ...........(ii)

where a is an appropriate positive constant.

(b) Substitute $y=2a{\sin }^{2}t,dy=4a\sin t\cos tdt$ in (ii) to derive the solution $x=a\left(2t-\sin 2t\right),y=a\left(1-\cos \theta \right)$ ..........(iii)

for which $t=y=0$ when $x=0$. Finally, the substitution of $\theta =2t$ : in (iii) yields the standard para metric equations $x=a\left(\theta -\sin \theta \right),y=a\left(1-\cos \theta \right)$ of the cycloid that is generated by a point on the rim of a circular wheel of radius a as it rolls along the x axis. [See Example 5 in Section 9Ăƒ of Edwards and Penney. Calculus: Early Transcendentals. 7th edition (Upper Saddle River, NJ: Prentice Hall. 2008).]