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The inverse cubic force

The problem of the inverse cubic force is simply the generalization of the inverse square force to the cubic case, ie a force of the form

\begin{equation} F \propto \frac{1}{r^3} \end{equation}

due to its very peculiar behaviour, it is quite an interesting problem to work through.

The classical case

\begin{equation} \vec{F} = k \frac{\vec{r}}{r^4} \end{equation}