I'm sorry for this homework question but I'm sitting a really long time now on this rather "easy" looking problem and I can't find a way to solve it.
I'm given the Hamiltonian of the particle: $$H=\frac{1}{2m}(p_r^2+\frac{p_{\theta}^2}{r^2}+\frac{p_{\phi}^2}{r^2\sin^2\theta})-\alpha Q^2\frac{e^{-\beta mr}}{r}.$$
I have to explicitly show that the z-component of the angular moment $L_z=mr^2\sin^2(\theta)\dot{\phi}=p_{\phi}$ is conserved by showing that $$\{L_z,H\}=0$$
I thought that this should be reltaive straightforward cause a lot of terms of the poisson bracket will be zero (e.g. $\frac{\partial L_z}{\partial \phi},\frac{\partial L_z}{\partial p_r},\frac{\partial L_z}{\partial p_{\theta}},\frac{\partial H}{\partial \phi}, $). Calculating the nonzero terms I get: $$\{L_z,H\}=2r\sin^2(\theta)\dot{\phi}p_r+2r\cos(\theta)\sin(\theta)\dot{\phi}p_{\theta}$$
Maybe it's obvious that this should be zero and I can't see it or I did a calculating error but if I did not I have to assume something in the problem description is wrong.
I would be awesome if anyone could help me out a little bit and maybe push me in the right direction. Thanks!