Inverse-square force

An inverse-square central force has magnitude proportional to the reciprocal square of the separation.

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 signed inverse-square force $\mathbf{F}=-\kappa \mathbf{r}/r^3$ on a particle of mass $m$, with Hamiltonian $H=p^2/(2m)-\kappa /r$, the Laplace-Runge-Lenz vector $\mathbf{A}=\mathbf{p}\times \mathbf{L}-m\kappa \hat{\mathbf{r}}$ is conserved. Attraction has $\kappa >0$ and repulsion has $\kappa <0$. It obeys $\mathbf{A}\cdot \mathbf{L}=0$ and $A^2=m^2{\kappa }^2+2mH{L}^2$. For unit mass in a Kepler orbit, $\kappa =GM$.

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