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:

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

which means:

H=\frac{p^2}{m}-\frac{p^2}{2m}+U

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: L = \frac{1}{2} m (\dot{r}^2+r^2 dot{\phi}^2)

p_r=m \dot{r} and p_{\phi}=mr^2 \phi

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

how would i implement theta into this?


And for cylindrical coordinates, i have this:

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


p_r=m \dot{r}
p_{\phi}=mr^2 \phi
p_{z}= zm

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

is this the right idea?