Solving a Coupled Oscillator Problem: A Puzzling Exercise

[wotanub]

Homework Statement

Just click the link, The image is huge, so I did not use IMG tags.
http://i.imgur.com/zWNRf.jpg

Homework Equations

Let's see, The rotational kinetic energy of a body is given as $K = \frac{1}{2}Iω^{2}$
for a point mass, $I = mr^{2}$
for a rigid rod rotating at it's end, $I = \frac{mL^{2}}{3}$

The Attempt at a Solution

http://i.imgur.com/qh2Fh.jpg

First, I'm trying to write the Lagrangian, but I'm not sure I got it quite right. I'm wary about that potential energy... My intuition says that's right, I was trying to write the component of the gravitational force in the direction of the angle(s).

Also, I'm not sure where the elongation of the string comes in, or even what "elongation" even really means. I thought x was the length the string changes according to the picture, so what is this ε?

And even after I finish writing the Lagrangian, I've never solved a coupled oscillator problem in terms or angles instead of displacements. How do normal modes come into play when the solutions to the equations of motion won't take the form of $q = Acos(ωt)$?

Any help is appreciated, this has got me scratching my head since my textbook has no examples even remotely similar.

 

[clamtrox]

First, I'm trying to write the Lagrangian, but I'm not sure I got it quite right. I'm wary about that potential energy... My intuition says that's right, I was trying to write the component of the gravitational force in the direction of the angle(s).

You're missing the rod / string length in the potential energy.

 

[wotanub]

For the rod, would I use a or 2a?

and the string, it should be (4a/3 + x), right?

Also I think I'm missing a term in both the kinetic and potential energy terms... Something to do with the elongation, but I'm not sure how to handle it.