Effective potential

An effective potential collects coordinate-dependent terms after other degrees of freedom have been eliminated or fixed by conservation laws. It allows the remaining coordinate to be studied as a simpler energy problem. The meaning depends on the reduction; an effective potential in quantum field theory is a different construction from the classical radial example below.

For a constant-mass particle in a central potential $U(r)$ with fixed angular momentum magnitude $L$, the conserved mechanical energy becomesAccessible radii satisfy $E\ge U_{\mathrm{eff}}$; equality gives a turning point when the radial motion reverses. A stationary point supports a circular orbit, and a strict local minimum gives radial stability in this reduced model. For $U=-k/r$ with $k>0$ and $L\ne 0$, differentiation gives $r_0=L^2/(mk)$ and $U_{\mathrm{eff}}^{\prime \prime }(r_0)=k/r_0^3>0$. Thus the centrifugal term can support a stable orbit even though the attractive potential alone has no minimum at positive $r$.