Hamiltonian in cylindrical coordinates

[Piano man]

Hi, I'm trying to find the Hamiltonian for a system using cylindrical coordinates.

I start of with the Lagrangian $$L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2)-U(r,\theta,z)$$

From that, using $$H=\sum p\dot{q}-L$$
$$=p_r\dot{r}+p_\theta\dot{\theta}+p_z\dot{z}-\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2)+U(r,\theta,z)$$
$$=\frac{1}{m}(p_r^2+p_\theta^2+p_z^2)-\frac{1}{2m}(p_r^2+p_\theta^2r^2+p_z^2)+U(r,\theta,z)$$
$$=\frac{1}{2m}[p_r^2+(2-r^2)p_\theta^2+p_z^2]+U(r,\theta,z)$$

But the standard answer is
$$H=\frac{1}{2m}(p_r^2+\frac{p_\theta^2}{r^2}+p_z^2)+U(r,\theta,z)$$

So where did I go wrong?

Thanks for any help :)

 

[kuruman]

What did you use for p_θ?

Don't forget that

$$p_j= \frac{\partial L}{\partial \dot{q}_j}.$$

 

[Piano man]

Ah, of course!
Kind of obvious really...


Thanks for the help :)