Example: Motion in a Central Potential $V(r)$

Use spherical coordinates and choose the $z$ direction so that the motion is in the $r\phi$ plane. The Hamiltonian is $T+U$ for this simple case if we don't choose coordinates that are time dependent. Due to the spherical symmetry of the potential (and of the kinetic energy term), the momentum conjugate to $\phi$ is conserved.

$$\dot{p}_i =-{\partial H\over\partial q_i }$$

$$\dot{p}_\phi =-{\partial H\over\partial \phi}=0$$

$$p_\phi = {\partial L\over \partial \dot{\phi}}={\partial T\over \partial \dot{\phi}}$$

$$T ={1\over 2}mv^2={1\over 2}m(\dot{r}^2+r^2\sin^2\theta\dot{\phi}^2+r^2\dot{\theta}^2) \rightarrow {m\over 2}(\dot{r}^2+r^2\dot{\phi}^2)$$

$$p_\theta = 0 = const.$$

$$p_\phi = m r^2\dot{\phi} = const.$$

$$p_r = m\dot{r}$$

$$H = {1\over 2m}(p_r^2+{p_\phi^2\over r^2}) + V(r)$$

$$\dot{p}_r =-{\partial H\over\partial r} =-{\partial \over\partial r}\left({p_\phi^2\over 2m r^2} + V(r)\right)$$

Either of the last two equations can be used to solve the problem using the effective potential.