Trying to show that a given forced vibration satisfies this equation of motion

1. Apr 24, 2010

quasar_4

1. The problem statement, all variables and given/known data

If the generalized driving forces Qi are not sinusoidal, and the dissipation function is simultaneously diagonalized along with T and V, show that the forced vibrations are given by $$\zeta_i = \frac{1}{2\pi}\int_{-\infty}^\infty \frac{G_i(\omega) (\omega_i^2 - \omega^2 + i \omega F_i) e^{-i\omega t}}{(\omega_i^2-\omega^2)^2 + \omega^2 F_i^2}dt$$

(This is basically the second half of Goldstein's chapter 6 exercise 15, 3rd edition, if that helps).

2. Relevant equations

When the dissipation function is simultaneously diagonalized with T and V, the normal coordinates decouple the equations of motion. This puts them in the form $$\ddot{\zeta_i} + F_i \dot{\zeta_i} + \omega_i^2 \zeta_i = 0$$.

Also, the Gi functions are defined to be the Fourier transforms of the generalized driving forces, Qi.

3. The attempt at a solution

I believe that we'll be done if we can show that the solution given for zeta satisfies the decoupled equations of motion. So, I tried taking some derivatives and putting it all together... but I can't get it to simplify to zero, and neither can Maple.

Actually, I don't understand why this is an integral over t, rather than omega. Part 1 of this problem had zetas as integrals over omega, and my solutions satisfied those equations of motion (those eqns of motion were for no damping, so now we've added damping)... then suddenly, zeta is an integral over t instead. If the Gi functions are Fourier transforms of the Qis, why would we then integrate them over t?

So, what I need to know is:
1- This isn't listed in Goldstein's textbook errata website, but should zeta be an integral over omega, or t?
2- assuming the integral over t is correct, is my method -- seeing if zeta satisifies the equations of motion -- rigorous enough? Any tips to simplify?
3 - The dissipation function is typically a function of time, right? I'd guess that this is so, although Goldstein doesn't explicitly say in the problem, because the generalized driving forces Qi are functions of time, and these are related in a simple way to the dissipation functions. If the dissipation functions are a function of time, this adds messiness to the derivatives...

Any help here would be greatly appreciated.