Central Force Fields

Central force fields are widely used in physics and they have simple yet important properties.

In general, central force is described using

\[\vec F(\vec r) = f(r)\hat r.\]

The Lagrangian for an object of mass \(m\) in a central force field is

\[\begin{split}L &= \frac{1}{2} m \mathbf{ \dot r} ^2 - V(r) \\ & = \frac{1}{2} m ( \mathbf{\dot r}^2 + r^2 \theta^2 ) - V(r) .\end{split}\]

The interesting thing for such a system is that there is always a conserved quantity since the Lagrangian has no explicit \(\theta\) dependence. It is obvious that

\[\frac{\partial L}{\partial \theta} = 0.\]

Now we have

\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) = 0,\]

which leads to the conservation of angular momentum as the first equation of motion,

\[\dot l \equiv \dot p_\theta = \frac{d}{dt} \left( m r^2 \dot\theta \right) = 0\]

The second equation of motion is given by

\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{r}} \right) - \frac{\partial L}{\partial r} = 0,\]

which simplifies to

\[\frac{d}{dt} (m \dot r) - m r {\dot \theta}^2 + \frac{\partial V(r)}{\partial r} = 0.\]

Applying the conserved quantity, we find an effective potential

\[V_{eff} (r) = V(r) + \frac{1}{2} \frac{ l^2 }{ m r^2 }.\]