 Solution

Because $H$ is independent of $\varphi$, Hamilton's equation gives ${\dot{p}}_{\varphi }=0$. PutUsingone obtains $d{L}^2/dt=0$. Physically, $p_{\varphi }=L_z$ is the component of angular momentum along the polar axis, while $L^2$ is the squared magnitude of the angular momentum.