Inverse-cube central force

A radial force proportional to $r^{-3}$. For potential $V=B/r^2$, its outward radial component is $2B/r^3$: positive $B$ is repulsive and negative $B$ is attractive.

For nonzero angular momentum $L$, put $u=1/r$ and $\lambda =1+2mB/L^2$. The Binet equation is $u_{\theta \theta }+\lambda u=0$. Positive $\lambda$ gives cosine scattering branches, zero gives linear reciprocal radius, and negative gives exponential combinations, with energy distinguishing sinh, exponential and cosh branches. Only intervals with $u>0$ are physical.

At $B=-L^2/(2m)$ the centrifugal and force potentials cancel. With zero radial speed, any radius is a circular solution, but there is no radial restoring force; radial-velocity perturbations are not confined.