 Solution

With $L=1/\sqrt{n^2-1}$, one has $\nu =n$. Since $n\ge 3>\sqrt{3}$, ii proves boundedness. Write the positive rational number $n=p/q$ in lowest terms. The radius and its angular derivative repeat after $\Delta \theta =2\pi /n$. After $p$ such radial oscillations,so both the position and the radial and transverse velocities return to their initial values. The elapsed time iswhich is finite and positive because $r$ is bounded above and below by positive constants. Thus the motion is periodic. The orbit is a closed, generally self-intersecting rosette orbit, with $p$ radial oscillations during $q$ complete revolutions; for integral $n$ it has $n$ evenly spaced lobes around the origin. It never reaches the origin and is not a circular orbit.