Example: Particle in 3D Potential

$\displaystyle L$	$\displaystyle = T-U= {1 \over 2} m(\dot{x}^2+\dot{y}^2+\dot{z}^2) - V(x,y,z)$
$\displaystyle p_x$	$\displaystyle = {\partial L\over \partial \dot{x}}= m\dot{x}$	we know these momenta
$\displaystyle p_y$	$\displaystyle = m\dot{y}$
$\displaystyle p_z$	$\displaystyle = m\dot{z}$
$\displaystyle H(q_i,p_i)$	$\displaystyle = \sum\limits_{i=1}^n p_i \dot{q}_i - L$	check $\displaystyle T+U$ this time
$\displaystyle H$	$\displaystyle = p_x\dot{x}+p_y\dot{y}+p_z\dot{z} -L$
$\displaystyle H$	$\displaystyle = {p_x^2+p_y^2+p_z^2\over 2m} + V(x,y,z) =T+U$
$\displaystyle \dot{q}_i$	$\displaystyle ={\partial H\over\partial p_i}$	gives us $\displaystyle p$ in terms of $\displaystyle v$
$\displaystyle \vec{v}$	$\displaystyle = {\vec{p}\over m}$
$\displaystyle \dot{p}_i$	$\displaystyle =-{\partial H\over\partial q_i }$	gives us equation of motion
$\displaystyle \dot{\vec{p}}$	$\displaystyle = -\vec{\nabla} V(x,y,z)$