Liouville's Theorem in Spherical Coordinates

Question

I'm attempting to verify the invariance of $$d\omega=\prod_{i=1}^{3N}dq_idp_i$$ in the case $N=1,$ and under a canonical transformation into spherical coordinates. I know to let $$q_1=r\sin\theta\cos\phi,\quad q_2=r\sin\theta\sin\phi\quad, q_3=r\cos\theta,$$ but what would be the appropriate transformation for the momenta? That is, how do you calculate the Jacobian in $$\int\prod_{i=1}^{3}dq_idp_i=\int J\prod_{i=1}^{3}dQ_idP_i$$ explicitly and find that $J=1$ as known from Liouville's theorem? I know that the Hamiltonian for a free particle in spherical coordinates is $$\mathcal{H}=\frac{1}{2m}\left(\dot{r}^2+\left(r\dot{\theta}\right)^2+\left(r\sin\theta\dot{\phi}\right)\right)=\frac{1}{2m}\left(p_r^2+\frac{p_{\theta}^2}{r^2}+\frac{p_{\phi}^2}{r^2\sin^2\theta}\right),$$ but I don't see how I could use this to say which $p_i$ is associated with which $P_i.$

Answer 1

You have written the Hamiltonian in terms of the new coordinates. The fact that

$ \frac{\partial}{\partial p_{\phi}} H = \frac{d \phi}{d t}$

from the Euler Lagrange equations is enough to tell you that the coordinates are conjugate in the Hamiltonian picture.