1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Homework Helper
    Gold Member

    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 :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Hamiltonian in cylindrical coordinates
Loading...