I need to find the Hamiltonian for a single particle under the influence of potential U in different coordinates:(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Homework Help: Hamiltonian for a single particle

Can you offer guidance or do you also need help?

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