Write down the Euler-Lagrange equation for the integral

$$\int f\left(y, y^{\prime}, x\right) d x$$

An ant is walking on the surface of a sphere, which is parameterised by $\theta \in[0, \pi]($ angle from top of sphere) and $\phi \in[0,2 \pi$ ) (azimuthal angle). The sphere is sticky towards the top and the bottom and so the ant's speed is proportional to $\sin \theta$. Show that the ant's fastest route between two points will be of the form

$$\sinh (A \phi+B)=\cot \theta$$

for some constants $A$ and $B$. $[A, B$ need not be determined.]