Central force problem

A central force always points along the line between a particle and a fixed centre, and its size depends only on distance:

$$\mathbf{F}(\mathbf{r})=F(r)\hat{\mathbf{r}}.$$

Examples include Newtonian gravity and the electrostatic force between point charges. The force has no sideways torque about the centre, so angular momentum is conserved and the motion stays in a plane.

Effective one-dimensional picture

With conserved angular momentum magnitude $\ell$, the radial motion can be treated using an effective potential:

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

The extra term is the centrifugal barrier. It is not a new force; it is what angular motion contributes when the problem is reduced to the radial coordinate.

Why it matters

The central-force problem is a meeting point for Newton’s Laws of Motion, Energy, Lagrangian mechanics, and Hamiltonian mechanics. It is the cleanest route into orbits, scattering, and the inverse-square laws used in gravity and electrostatics.

Common pitfalls

• Central means directed toward or away from a centre, not necessarily attractive.
• Conserved angular momentum follows from zero torque, not from circular motion.
• The radial coordinate can change even when angular momentum is fixed.

Source trail: Susskind The Theoretical Minimum index; reference book: Classical Mechanics: The Theoretical Minimum by Leonard Susskind and George Hrabovsky.