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, $\kappa =GM$.

For $H=p^2/2-\kappa /r$, the angular momentum and Laplace-Runge-Lenz vector obey $\{K_i,K_j\}=-2H{ϵ}_{ijk}{L}_k$. On negative or positive energy regions use $K/\sqrt{|2H|}$ to obtain the compact or Lorentz Lie algebra, respectively. At zero energy the algebra on the characteristic orbit quotient is the Euclidean motion algebra. Global group actions require complete generators or collision regularization; simply replacing $H$ by a constant inside a Hamiltonian-vector-field derivative is not valid on the unreduced energy hypersurface.

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