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Hamiltonian for a single particle

  1. Oct 21, 2006 #1
    I need to find the Hamiltonian for a single particle under the influence of potential U in different coordinates:

    I have found the Hamiltonian for Cartesian coordinates fairly easily and would just like a check if it is:

    [tex]L=\frac{1}{2} m \dot{q}^2 -U[/tex] with [tex]p=m \dot{q}[/tex]

    which means:

    [tex]H=\frac{p^2}{m}-\frac{p^2}{2m}+U[/tex]

    I have tried spherical but I cannot implement theta, I tried it in two-d but do not know how to get the Lagrangian in using r,phi,and theta.

    I know that: [tex] L = \frac{1}{2} m (\dot{r}^2+r^2 dot{\phi}^2) [/tex]

    [tex]p_r=m \dot{r}[/tex] and [tex]p_{\phi}=mr^2 \phi[/tex]

    So that this means: [tex] H = \frac{p_r ^2}{2m}+\frac{p_{\phi} ^2}{2 m r^2}+U[/tex]

    how would i implement theta into this?


    And for cylindrical coordinates, i have this:

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


    [tex]p_r=m \dot{r}[/tex]
    [tex]p_{\phi}=mr^2 \phi[/tex]
    [tex]p_{z}= zm[/tex]

    so that [tex]H=\frac{p_{r} ^2}{2m}+\frac{p_{\phi} ^2}{2mr^2}+ \frac{p_{z} ^2}{2m}[/tex]

    is this the right idea?
     
  2. jcsd
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