# 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

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

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

Now we have

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

The second equation of motion is given by

which simplifies to

Applying the conserved quantity, we find an effective potential