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 behavior, it is quite an interesting problem to work through.

The classical case

The classical case is given by the obvious force,

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

where $k$ is some proportionality constant, generally taken to be positive. If we take this problem as a central force problem, and we shift the coordinates so that the center is at coordinates $0$, we therefore have some Newtonian equation of the form

\begin{equation} \ddot{x}(t) = k \frac{\vec{x}}{|\vec{x}|^4} \end{equation}