Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 16B / b / Solution

The equations areHencewhich is not generally zero. Conservation requires $\beta /\alpha =1$. Then the potential is $\alpha (x^2+y^2{)}^2/4$, so the action has rotational symmetry. By Noether's theorem, $J$ is the conserved angular momentum, up to the sign convention used in its definition.