Green's function for a critically damped oscillator

[shyta]

Homework Statement

Consider critically damped harmonic oscillator, driven by a force F(t)
Find the green's function G(t,t') such that x(t) = ∫ dt' G(t,t')F(t') from 0 to T solves the equation of motion with x(0) =0 and x(T) =0

Homework Equations

x(t) = ∫ dt' G(t,t')F(t') from 0 to T

The Attempt at a Solution

Hi guys, I am completely new to green's function.. need a lot of help understanding the use, and how to use it >_<

I've been doing some readings and this is what i understand so far

x = $x _{h}$ + ∫G(t,t')f(t') dt'

i.e. G(t,t') will be the particular solution to the ode with F(t) = δ(t-t')

does this mean that I should let $\ddot{x}$ +2γ$\dot{x}$ + $ω _{0}$² = δ(t-t') and solve this to get the green's function?

 

[Hootenanny]

does this mean that I should let $\ddot{x}$ +2γ$\dot{x}$ + $ω _{0}$² = δ(t-t') and solve this to get the green's function?

Yes. Except that there's a small typo in your expression. It should be

$$\ddot{x} +2\gamma\dot{x} + \omega_{0}^2 x = \delta(t-t^\prime)$$

A nice way to solve the above equation is to write the RHS as the inverse of a Fourier transform, meaning the RHS becomes a harmonic function of time.

 

[shyta]

Hello! thanks for you reply! :)

yeah you are right, i missed out the x there

not sure if i got it right but on the RHS, i got $\frac{1}{2\pi}$$\frac{1}{i(t-t')}$$e^{i\omega(t-t')}$+C

is this right?

oh and (d²/dt² + 2γd/dt + ω0²) G(t,t') = δ(t-t')
so i should be solving as usual 2nd order ode to get my answer for homogeneous and inhomo am i doing the steps right?

 

[mathfeel]

Hello! thanks for you reply! :)

yeah you are right, i missed out the x there

not sure if i got it right but on the RHS, i got $\frac{1}{2\pi}$$\frac{1}{i(t-t')}$$e^{i\omega(t-t')}$+C

is this right?

oh and (d²/dt² + 2γd/dt + ω0²) G(t,t') = δ(t-t')
so i should be solving as usual 2nd order ode to get my answer for homogeneous and inhomo am i doing the steps right?

Try taking the Fourier transform of both sides. To makes things simpler, recognize that the dependent of t and t' in G has this form: $G(t, t^\prime) = G(t - t^\prime)$

 

[shyta]

im really bad at Fourier transform, so i didnt follow your advice..

but i did try considering the force being written as a Fourier transform
f(t) = 1/2$\pi$∫F(w)$e^{iwt}$ dw
and the dirac delta
δ(t-t')= 1/2$\pi$∫ $e^{iwt}$$e^{-iwt'}$ dw

so i went ahead to solve and i got that

G(t,t') = $x_{h}$ + 1/2$\pi$ ∫ ($e^{iwt}$$e^{-iwt'}$) / ($-\omega^{2}$ + 2i$\gamma\omega$ + $\omega_{0}^{2}$) dw



and I am stuck at this integration lol

 

[Hootenanny]

im really bad at Fourier transform, so i didnt follow your advice..

but i did try considering the force being written as a Fourier transform
f(t) = 1/2$\pi$∫F(w)$e^{iwt}$ dw
and the dirac delta
δ(t-t')= 1/2$\pi$∫ $e^{iwt}$$e^{-iwt'}$ dw

so i went ahead to solve and i got that

G(t,t') = $x_{h}$ + 1/2$\pi$ ∫ ($e^{iwt}$$e^{-iwt'}$) / ($-\omega^{2}$ + 2i$\gamma\omega$ + $\omega_{0}^{2}$) dw
and I am stuck at this integration lol

The integration is a little tricky. You're going to need to use complex analysis, which I'm assuming you've at least met. Since we are dealing with a critically damped oscillator, we can set $\gamma=1$. What we're going to do now is extend the integral into the complex plane. Let $C\subset\mathbb{C}$ be a semi-circle in the upper half plane, that is, $C=\{\omega:|\omega|< a, \omega\geq0\}$. So, we have

$$I=\int_C \frac{e^{i\omega(t-t^\prime)}}{-\omega^2 + 2i\omega_0\omega + \omega_0^2}\;\text{d}\omega$$

Can you do the next step, maybe using Cauchy's Residue Theorem?

 

[shyta]

Hmm nope I have never heard of the residue theorem. I tried wiki-ing it to see how it works, but it looks complicated. Is there any other way to do this?

 

[Hootenanny]

Hmm nope I have never heard of the residue theorem. I tried wiki-ing it to see how it works, but it looks complicated. Is there any other way to do this?

I can't immediately see another straightforward method to evaluate the integral. Let me ask a few of the other homework helpers and get back to you.

 

[vela]

The integration is a little tricky. You're going to need to use complex analysis, which I'm assuming you've at least met. Since we are dealing with a critically damped oscillator, we can set $\gamma=1$.

The differential equation for the damped harmonic oscillator is usually
$$\ddot{x} + 2\zeta\omega_0 \dot{x} + \omega_0^2x = 0$$There's a factor of $\omega_0$ in the damping term, and with this convention, a critically damped oscillator would correspond to $\zeta=1$, so I think you actually want $\zeta=\omega_0$ here.

 

[Hootenanny]

The differential equation for the damped harmonic oscillator is usually
$$\ddot{x} + 2\zeta\omega_0 \dot{x} + \omega_0^2x = 0$$There's a factor of $\omega_0$ in the damping term, and with this convention, a critically damped oscillator would correspond to $\zeta=1$, so I think you actually want $\zeta=\omega_0$ here.

Good catch! I didn't notice that :redfaced:

I will correct my earlier posts.

 

[vela]

Another way to find the Green's function is to solve the differential equation in the two regions 0≤t<t' and t'<t≤T. Apply the boundary conditions, and then match the two solutions at t=t'. The Green's function needs to be continuous at the boundary, but its derivative is not. By integrating the differential equation from t'-ε to t'+ε and letting ε→0, you can find the discontinuity in the derivative.