Frequency of small oscillations

[LHarriger]

Does anyone know where I can get some information on how you can relate the frequency of small oscillations to the second derivative of potential energy. I saw this done recently in a qualifying exam level problem but I do not remember learning this method and it is not in my classical dynamics book. See below if you want a more extensive context for this question.

I solved a problem recently where you were given two masses m and M connected by a string. The first mass was set rotating on a frictionless table. The string passed through a hole in the center of the table allowing the second mass to hang vertically under gravity. I was asked to:

1) Set up the Langrangian and derive eqns of motion.
2) Show that the orbit is stable with respect to small changes in orbit.
3) Find the frequency of small oscillations.

I was able to do the first two without any problem but got stuck on the third. The d.e. was too messy to solve by hand in order to acquire the freqency. I looked at the solution and they used the approximation:
\omega^{2}=\frac{1}{M_{eff}}\frac{ \partial^{2}U_{eff}}{\partial r^{2}}\mid_{r=r_0}}
where ro is the stable point and Meff = M+m
Where can I get more information discussing this approximation method?

 

[AlephZero]

If it's any help, for a simple spring and mass system

PE = (1/2)Kx^2

d^2(PE)/dx^2 = K

And w^2 = K/M.

So the same result would follow for small (linearized) oscillations of any system I suppose, if you expanded the PE and KE as Taylor series to get the effective mass and stiffness of the system about the equilibrium condition.