Central Forces

A central force field is one whose action is always directed toward a fixed point. If that fixed point is taken to be the origin, and if i is taken to be a unit vector directed from the origin to the position r of a particle acted upon by the central force, then the force field can be written 83 $$F(r)=-mP(r)i,$$ where P is a scalar function and m is the mass of the particle. The equation of motion for a particle of constant mass can be written down using Newton’s Second Law; it is 84 $$\ddot{r}=-Pi.$$