Green's Function for Newtons Force Equation

[JesseC]

This isn't so much a problem as a step in some maths that I don't understand: (I'm trying to follow a very badly written help sheet)

Here's how it goes:

Given Newtons equation

$$m \ddot{x} = F$$

The Green's function for this equation is given by

$$\ddot{G}(t,t^\prime)=\delta(t-t^\prime)$$ (1)

With initial conditions

$$G(t_1,\acute{t})=G(t_2,t^\prime)=0$$

We obtain the Green's function for t not equal to t':

$$G=A(t-t_1), \ t<t^\prime$$
$$G=B(t-t_2), \ t>t^\prime$$ (2)

Now I'm new to greens functions so I don't understand how you get from (1) to (2), and why are there two solutions for different domains. Are we guessing solutions of the form given in (2) and seeing if it works or is it just 'obvious' and I'm not seeing why. Thanks to anyone who can help me understand!

 

[vela]

The delta function splits the interval [t₁,t₂] into two pieces at t=t'. The idea is to solve the differential equation in each region and then match them up at the boundary t=t'.

When t≠t', δ(t-t')=0, so you're looking for solutions to

$$\ddot{G(t)} = 0$$

in each region. When you impose the initial conditions G(t₁)=G(t₂)=0, you get the two solutions given, where A and B are arbitrary constants.

The next step would be to require G(t) be continuous at t=t', and that will give you a relationship between A and B.

 

[JesseC]

Thanks, I understand that now!