Example: A spherical pendulum

Consider a pendulum bob of mass $\bgroup\color{black}$ m$\egroup$ hanging from the ceiling by a string of length $\bgroup\color{black}$ \ell$\egroup$ and free to move in two dimensions like the Foucault pendulum. The free variables are $\bgroup\color{black}$ \theta$\egroup$ and $\bgroup\color{black}$ \phi$\egroup$ of spherical coordinates and the energies are given by

$\displaystyle U$	$\displaystyle =-mgl\cos\theta$
$\displaystyle T$	$\displaystyle ={1\over 2}m\ell^2\left(\dot{\theta}^2+\sin^2\theta\dot{\phi}^2\right)$
$\displaystyle H$	$\displaystyle =T+U .$

$\displaystyle p_\phi$	$\displaystyle = {\partial T\over \partial \dot{\phi}}=m\ell^2\sin^2\theta\dot{\phi}$
$\displaystyle p_\theta$	$\displaystyle = {\partial T\over \partial \dot{\theta}}=m\ell^2\dot{\theta}$
$\displaystyle H$	$\displaystyle =T+U={1\over 2m\ell^2}\left(p_\theta^2+{p_\phi^2\over \sin^2\theta}\right) -mgl\cos\theta$
$\displaystyle \dot{p}_\phi$	$\displaystyle =-{\partial H\over\partial \phi}=0$
$\displaystyle \dot{p}_\theta$	$\displaystyle =-{\partial H\over\partial \theta}={p_\phi^2\cos\theta\over m\ell^2\sin^3\theta}-mg\ell \sin\theta$