Chapter 6

Consider a ray of light traveling in a vacuum from point $P_1$ to $P_2$ by way of the point $Q$ on a plane mirror, as in Figure $6.8.$ Show that Fermat's principle implies that, on the actual path followed, $Q$ lies in the same vertical plane as $P_1$ and $P_2$ and obeys the law of reflection, that ${\theta }_1={\theta }_2.[$ Hints: Let the mirror lie in the $xz$ plane, and let $P_1$ lie on the $y$ axis at $(0,y_1,0)$ and $P_2$ in the $xy$ plane at $(x_2,y_2,0)$ Finally let $Q=(x,0,z)$. Calculate the time for the light to traverse the path $P_{1}Q{P}_2$ and show that it is minimum when $Q$ has $z=0$ and satisfies the law of reflection.]

A ray of light travels from point $P_1$ in a medium of refractive index $n_1$ to $P_2$ in a medium of index $n_2,$ by way of the point $Q$ on the plane interface between the two media, as in Figure $6.9.$ Show that Fermat's principle implies that, on the actual path followed, $Q$ lies in the same vertical plane as $P_1$ and $P_2$ and obeys Snell's law, that $n_1\sin {\theta }_1=n_2\sin {\theta }_2.$ [Hints: Let the interface be the $xz$ plane, and let $P_1$ lie on the $y$ axis at $(0,h_1,0)$ and $P_2$ in the $x,y$ plane at $(x_2,-h_2,0).$ Finally let $Q=(x,0,z)$ Calculate the time for the light to traverse the path $P_{1}Q{P}_2$ and show that it is minimum when $Q$ has $z=0$ and satisfies Snell's law.]

Consider a right circular cylinder of radius $R$ centered on the $z$ axis. Find the equation giving $\varphi$ as a function of $z$ for the geodesic (shortest path) on the cylinder between two points with cylindrical polar coordinates $(R,{\varphi }_1,z_1)$ and $(R,{\varphi }_2,z_2).$ Describe the geodesic. Is it unique? By imagining the surface of the cylinder unwrapped and laid out flat, explain why the geodesic has the form it does.

Find the equation of the path joining the origin $O$ to the point $P(1,1)$ in the $xy$ plane that makes the integral ${\int }_o^P(y^2+y{y}^{\prime }+y^2)dx$ stationary.

Find and describe the path $y=y(x)$ for which the integral ${\int }_{x_1}^{x_2}\sqrt{x}\sqrt{1+y^{\prime 2}}dx$ is stationary.

Find the geodesics on the cone whose equation in cylindrical polar coordinates is $z=\lambda \rho .$ [Let the required curve have the form $\varphi =\varphi (\rho ).$ ] Check your result for the case that $\lambda \to 0$.

Show that the shortest path between two given points in a plane is a straight line, using plane polar coordinates.

An aircraft whose airspeed is $v_{\mathrm{o}}$ has to fly from town $O$ (at the origin) to town $P$, which is a distance $D$ due east. There is a steady gentle wind shear, such that ${\mathbf{v}}_{\text{wind }}=Vy\hat{\mathbf{x}},$ where $x$ and $y$ are measured east and north respectively. Find the path, $y=y(x),$ which the plane should follow to minimize its flight time, as follows: (a) Find the plane's ground speed in terms of $v_o,V,\varphi$ (the angle by which the plane heads to the north of east), and the plane's position. (b) Write down the time of flight as an integral of the form ${\int }_0^{D}fdx.$ Show that if we assume that $y^{\prime }$ and $\varphi$ both remain small (as is certainly reasonable if the wind speed is not too large), then the integrand $f$ takes the approximate form $f=(1+\frac{1}{2}{y}^2)/(1+ky)$ (times an uninteresting constant) where $k=V/v_{\mathrm{o}}$. (c) Write down the Euler-Lagrange equation that determines the best path. To solve it, make the intelligent guess that $y(x)=\lambda x(D-x),$ which clearly passes through the two towns. Show that it satisfies the EulerLagrange equation, provided $\lambda =(\sqrt{4+2k^2{D}^2}-2)/\left(k{D}^2\right).$ How far north does this path take the plane, if $D=2000$ miles, $v_{\mathrm{o}}=500\mathrm{mph},$ and the wind shear is $V=0.5\mathrm{mph}/\mathrm{mi}?$ How much time does the plane save by following this path? [You'll probably want to use a computer to do this integral.]

Consider a medium in which the refractive index $n$ is inversely proportional to $r^2$; that is, $n=a/r^2,$ where $r$ is the distance from the origin. Use Fermat's principle, that the integral (6.3) is stationary, to find the path of a ray of light travelling in a plane containing the origin. [Hint: Use twodimensional polar coordinates and write the path as $\varphi =\varphi (r).$ The Fermat integral should have the form $\int f(\varphi ,{\varphi }^{\prime },r)dr,$ where $f(\varphi ,{\varphi }^{\prime },r)$ is actually independent of $\varphi .$ The Euler-Lagrange equation therefore reduces to $\partial f/\partial {\varphi }^{\prime }=$ const. You can solve this for ${\varphi }^{\prime }$ and then integrate to give $\varphi$ as a function of $r.$ Rewrite this to give $r$ as a function of $\varphi$ and show that the resulting path is a circle through the origin. Discuss the progress of the light around the circle.]