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Homework Help: Hamiltonian in cylindrical coordinates

  1. Aug 17, 2010 #1
    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 :)
     
  2. jcsd
  3. Aug 17, 2010 #2

    kuruman

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    What did you use for pθ?

    Don't forget that

    [tex]p_j= \frac{\partial L}{\partial \dot{q}_j}.[/tex]
     
  4. Aug 17, 2010 #3
    Ah, of course!
    Kind of obvious really...
    :redface:

    Thanks for the help :)
     
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