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 [tex] L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2)-U(r,\theta,z) [/tex]

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

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

So where did I go wrong?

Thanks for any help :)

 

[kuruman]

What did you use for p_θ?

Don't forget that

[tex]p_j= \frac{\partial L}{\partial \dot{q}_j}.[/tex]

 

[Piano man]

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


Thanks for the help :)