Diagonalize a coupled damped driven oscillator

[jamie.j1989]

Homework Statement

I am trying to follow a paper, https://arxiv.org/pdf/1410.0710v1.pdf, I want to get the results obtained in equations 5 and 6 but can't quite work out how eq 3 has been diagonalized.

Homework Equations

eq 3

The Attempt at a Solution

As the system is driven i thought I'd try a solution first and then try to rearrange into an eigenvalue problem. I tried the solution $x=Ae^{i\omega t}$ where $\omega$ is the driving frequency and A some constant, I tried this solution as the system will oscillate at the driving frequency?

 

[Dr Transport]

I think that eq 3 (in the [ ] at the beginning) should be $\frac{d^2}{dt^2} + \gamma \frac{d}{dt} + \Omega^2 _0$

 

[jamie.j1989]

I didn't even notice that mistake and have been reading it as you have corrected. Would you suggest trying a solution and rearranging is the best method?

 

[Dr Transport]

on further review, maybe they don't treat the damping initially, $\gamma$ doesn't appear in the diagonalized matrix...

 

[jamie.j1989]

Yes you're right, if i don't consider damping I get the correct expressions for the eigenvalues, why would they not include the damping term? Also I'm struggling to understand the transformation matrix U, Is this a rotating frame transformation?

 

[Dr Transport]

$U$ allows for a normal mode analysis..., thus decoupling the equations of motion.