|m = - mx(t) + F(t) .||(1)|
The physical situation generally has F(t) applied for a finite interval. That is
|F(t) = .||(2)|
|x = x .||(3)|
We define the Green's function to be a solution to the adjoint equation with a delta function (i.e. impulsive) force. The adjoint equation will be identical in this case, but in general odd time derivatives corresponding to frictional forces will have their signs changed by the integration by parts needed to define the adjoint.
Our Green's function therefore satisfies
|m = - mG(t',t) + (t - t') .||(4)|
Just like the original equation of motion, we need to have boundary conditions on G(t',t) . In this case, because the Green's function itself is not the physical solution we seek, we are free to pick boundary conditions. As usual, some choices lead to simpler equations for particular x(t) boundary conditions than other choices. Two standard choices lead to the so-called retarded and advanced Green's functions.
Before talking about those boundary conditions, we need to look at the form of the Green's function equation. As noted above, it is the solution of the adjoint equation in the second argument for the delta function source at the right hand argument. For the special case where the coefficients of the linear equation are independent of time, the adjoint equation for the right hand argument is the same as the original equation for the left hand argument. That is, because the only dependence on t is in the derivatives, you can change variables from t to t - t' , with no change in the form of the equation, and then to t' which changes the sign of the odd time derivatives. These are exactly the sign changes needed to change the adjoint equation to the original equation. Therefore for these special cases, the Green's function is also the solution of the original equation on the left hand variable with an impulsive force at the time corresponding to the right hand variable.
The retarded Green's function is the solution where the oscillator is at rest at the origin before the impulsive force is applied, and therefore after the force, the oscillator will be vibrating. The advanced Green's function is the solution where the oscillator is at rest at the origin after the impulsive force is applied. Therefore it is vibrating at exactly the right amplitude and phase before the force so that the force cancels all the motion. Both solutions are valid and answer different physical questions. The retarded Green's function tells you what the amplitude and phase of the oscillator, initially at rest at the origin, will be after an impulsive force is applied. The advanced Green's function tells you the amplitude and phase the oscillator would need initially so that it would end up at rest at the origin after the impulsive force is applied. While it is true that we ask the first question more often than the second, both are equally physical questions, and neither violates causality. Note also that these conditions are reversed for the adjoint equation which is the one we really should solve. That is, the retarded Green's function is the solution of the adjoint equation that has the oscillator at rest at the origin after the impulse, while the advanced Green's function is the solution of the adjoint equation that has the oscillator at rest at the origin before the impulse.