Hamiltonian for a single particle

[UrbanXrisis]

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?

 

[Jhero]

Hello,

Thank you for your post. The Hamiltonian for a single particle under the influence of potential U in different coordinates is indeed different for each coordinate system. For Cartesian coordinates, your Hamiltonian is correct. However, for spherical coordinates, the correct Hamiltonian is:

H = \frac{p_r^2}{2m} + \frac{p_{\phi}^2}{2mr^2} + \frac{p_{\theta}^2}{2mr^2 \sin^2(\phi)} + U

Here, p_{\theta} is the momentum conjugate to \theta. To derive this Hamiltonian, you can use the transformation equations:

p_r = m\dot{r}
p_{\phi} = mr^2\dot{\phi}
p_{\theta} = mr^2\sin^2(\phi)\dot{\theta}

For cylindrical coordinates, your Hamiltonian is also correct. However, it is important to note that the momentum conjugate to z should be p_z, not p_{z}. So the correct Hamiltonian is:

H = \frac{p_r^2}{2m} + \frac{p_{\phi}^2}{2mr^2} + \frac{p_z^2}{2m} + U

I hope this helps clarify the Hamiltonian for different coordinate systems. Let me know if you have any further questions. Good luck with your research!