Repulsive inverse-square force

The repulsive inverse-square force $\mathbf{F}=k\mathbf{r}/r^3$ has potential energy $V(r)=k/r$ and scatters positive-energy trajectories along hyperbolic conics.

For the repulsive inverse-square Hamiltonian $H={\mathbf{p}}^2/(2m)+k/r$, the Laplace-Runge-Lenz vectoris conserved. It obeys $\mathbf{A}\cdot \mathbf{L}=0$ and $A^2=m^2{k}^2+2mH{L}^2$.

Taking the dot product of the Laplace-Runge-Lenz vector with $\hat{\mathbf{r}}$ gives $A\cos \theta =L^2/r+mk$. Thus