Difficulties understanding Green's Functions

[harsh]

Hello people.
I am trying to understand how the Green's functions work, and how to come up with one for a given differential equation. Now, I need to write down the Green's function for 3 different types of differential equations. They are simply, underdamped, critically damped, and overdamped oscillators. I know that the Green's functions are easy to come up with for above mentioned cases if you know the homogenous solution (just replace t with (t - t')), but I am not exactly sure how the heavyside (step-function) gets involved with this. Moreover, I know that the Dirac-delta function is very useful in solving the nasty integrals, but I don't really understand how we are using them. I basically need help in coming up with Green's functions for 3 different kinds of oscillators. Any help on the step-function and the impulse function (as forcing functions), would also be greatly appreciated. Thanks in advance for any help.

- harsh

 

[Dr Transport]

First of all, the three cases you list are for different values of the damping constant for an oscillator differential equation.

To find the correct Greens' function, you need to do two things
1. Solve the differential equation

$$\frac{d^{2}}{dx^{2}} G(x) - \gamma\frac{d}{dx} G(x) - k{^2} G(x) = -\delta{x}$$

subject to the bondary conditions that you are given. The other equation relates continuity of the derivative of the Greens' function at $$x = 0$$. Now for the case of the undamped harmonic oscillator, this is

$$\frac{d}{dx}G(0^{-}) - \frac{d}{dx}G(0^{+}) = -1$$ for the damped oscillator will be different, but you get the idea. Then you can get the Greens' function, you can not just find the solution and put $$t - t'$$ in and call it quits. The solution is more difficult than that. If you want time dependence, you follow what I have quickly outlined and work in the time the difference is that you need two $$\delta$$ funtions, one in space and one in time.

Have fun and play around with it and I'll do the same.

dt