Find the period of radial oscillation through effective potentials

[Richardbryant]

Homework Statement

Given circuit is a circle, force is a central force[/B]
Ueff(r)=U(r)+L^2/2mr^2

Homework Equations

the problem i find is, the angular momentum is a function of r
however, the solution when differentiate the effective potential, just treat angular momentum as a constant.
That's the point i am puzzle of, what is the physical sense of treating angular momentum as a constant? isn't it depends on the position r?

The Attempt at a Solution

 

[mfb]

Angular momentum is conserved if there are only radial forces - the particle can change its radius, that is no problem.