Central Force Motion

Setup

For a central potential $V(r)$, the Lagrangian in polar coordinates is

$$L=\frac{1}{2}m(\dot r^2+r^2\dot\theta^2)-V(r).$$

Angular Momentum

The coordinate $\theta$ is cyclic, so

$$\ell = \frac{\partial L}{\partial \dot\theta}=mr^2\dot\theta$$

is conserved.

Effective Potential

The radial motion can be written as one-dimensional motion in

$$V_{\rm eff}(r)=V(r)+\frac{\ell^2}{2mr^2}.$$

The second term is the centrifugal barrier.