# So close to a perfect circle

**Classical Mechanics**Level 5

Consider the motion of a point mass under the influence of an attractive central force. The potential for the force is proportional to the inverse distance from the central point O to the point mass. For every value of the angular momentum \(L\) of the object around O we have an equilibrium point in the radial direction, which corresponds to a circular orbit. We now consider a small perturbation of a circular orbit into a low eccentricity ellipse with the same angular momentum. If we look at the radial distance from O to the object on the elliptical orbit, we see that the distance changes from some \(r_{min}\) to \(r_{max}\) with some frequency \(f\). If at \(L_1=1~\mbox{kg}\cdot\mbox{m}^2/\mbox{s}\) we get \(f_1=f(L_1)=1~\mbox{Hz}\), find the frequency \(f_2\) for \(L_2=10~\mbox{kg}\cdot\mbox{m}^2/\mbox{s}\) **in Hz**.

Bonus thought: The value of the frequency at \(L_3=0\) (without angular momentum) is defined as the bare frequency. Is this a finite value?