| course |
course_year |
question_number |
tags |
title |
year |
Variational Principles |
IB |
79 |
IB |
2014 |
Variational Principles |
|
Paper 2, Section II, C |
2014 |
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.]