Repulsive inverse-square potential ($V(r)=c/r^2,c>0$)

This potential energy generates a repulsive central force $F_r=-V^{\prime }(r)=2c/r^3$. It is an inverse-square potential but an inverse-cube force; the powers must not be confused with the Coulomb or gravitational potential proportional to $1/r$.

For $V(r)=c/r^2$ with $c>0$, nonzero physical angular momentum $h$ and positive energy $E$, put $\beta =\sqrt{1+2mc/h^2}>1$ and $r_{\min }=\sqrt{(c+h^2/(2m))/E}$. Direct integration of the effective potential radial equation gives the displayed orbit. It has one radial minimum and two straight-line asymptotes separated by polar angle $\pi /\beta$. The repulsive scattering deflection is $\pi -\pi /\beta$; no bounded or circular orbit exists at finite positive energy.