A function $\theta(\phi)$ with given values of $\theta\left(\phi_{1}\right)$ and $\theta\left(\phi_{2}\right)$ makes the integral

$$I=\int_{\phi_{1}}^{\phi_{2}} \mathcal{L}\left(\theta, \theta^{\prime}\right) d \phi$$

stationary with respect to small variations of $\theta$ which vanish at $\phi_{1}$ and $\phi_{2}$. Show that $\theta(\phi)$ satisfies the first integral of the Euler-Lagrange equation,

$$\mathcal{L}\left(\theta, \theta^{\prime}\right)-\theta^{\prime}\left(\partial \mathcal{L} / \partial \theta^{\prime}\right)=C,$$

for some constant $C$. You may state the Euler-Lagrange equation without proof.

It is desired to tow an iceberg across open ocean from a point on the Antarctic coast (longitude $\phi_{1}$ ) to a place in Australia (longitude $\phi_{2}$ ), to provide fresh water for irrigation. The iceberg will melt at a rate proportional to the difference between its temperature (the constant $T_{0}$, measured in degrees Celsius and therefore negative) and the sea temperature $T(\theta)>T_{0}$, where $\theta$ is the colatitude (the usual spherical polar coordinate $\left.\theta\right)$. Assume that the iceberg is towed at a constant speed along a path $\theta(\phi)$, where $\phi$ is the longitude. Given that the infinitesimal arc length on the unit sphere is $\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)^{1 / 2}$, show that the total ice melted along the path from $\phi_{1}$ to $\phi_{2}$ is proportional to

$$I=\int_{\phi_{1}}^{\phi_{2}}\left(T(\theta)-T_{0}\right)\left(\theta^{\prime 2}+\sin ^{2} \theta\right)^{1 / 2} d \phi$$

Now suppose that in the relevant latitudes, the sea temperature may be approximated by $T(\theta)=T_{0}(1+3 \tan \theta)$. (Note that $(1+3 \tan \theta)$ is negative in the relevant latitudes.) Show that any smooth path $\theta(\phi)$ which minimizes the total ice melted must satisfy

$$\theta^{\prime 2}=\sin ^{2} \theta\left(\frac{1}{4} k^{2} \tan ^{2} \theta \sin ^{2} \theta-1\right),$$

and hence that

$$\sin ^{2} \theta=\frac{2}{1-\left(1+k^{2}\right)^{1 / 2} \sin 2\left(\phi-\phi_{0}\right)},$$

where $k$ and $\phi_{0}$ are constants.

[Hint:

$$\int \frac{d x}{x\left(\alpha^{2} x^{4}+x^{2}-1\right)^{1 / 2}}=\frac{1}{2} \arcsin \left[\frac{x^{2}-2}{x^{2}\left(1+4 \alpha^{2}\right)^{1 / 2}}\right]$$