Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 9E / b / Solution

Write $H=F(P,Q)$ with $P=\sum p_k^2$ and $Q=\sum q_k^2$. The angular momentum component $L_i={ϵ}_{ijk}{q}_j{p}_k$ has ${\partial }_{p_{\ell }}{L}_i={ϵ}_{ij\ell }{q}_j$ and ${\partial }_{q_{\ell }}{L}_i={ϵ}_{i\ell k}{p}_k$. Thereforesince a symmetric product contracts to zero with the antisymmetric Levi-Civita symbol. All three components of angular momentum are conserved by this rotationally invariant Hamiltonian.