What is the Green function solution for a damped harmonic oscillator?

[WiFO215]

I was reading up on Classical Mechanics and the general method for solving for an undamped harmonic oscillator was given as

[tex]\frac{d^{2}q}{dt^{2}} + \omega^{2}q = F(t)[/tex]

was solved using the Green function, G, to the equation

[tex]\frac{d^{2}G}{dt^{2}} + \omega^{2}q = \delta(t-t')[/tex]

and then integrating via the normal procedure.

Then the next case considered was the damped harmonic oscillator which had a damping term proportional to [tex]\frac{\omega}{Q}[/tex] times the velocity. The equation has the form

[tex]\frac{d^{2}q}{dt^{2}} + \frac{\omega}{Q}\frac{dq}{dt} + \omega^{2}q = F(t)[/tex]

Now, the author wants to find the particular solution to this equation and says that his Green function is of the form

G = [tex] A e^{\frac{{\omega(t-t')}}{2Q} + i\omega(t-t')}[/tex] plus this term's complex conjugate. This is a solution to STEADY STATE ONLY but does not consider the transient part.

BUT, he also claims that the above Green function is a solution the equation

[tex]\frac{d^{2}G}{dt^{2}} + \frac{\omega}{Q}\frac{dG}{dt} + \omega^{2}G = \delta(t-t')[/tex]

How can he do this? The Green function which will solve the above equation will also have a transient part apart from the steady state function which he has considered. G will be of the form above PLUS a dying out part. What happens to that?

 

[jvicens]

Thank you for bringing up this interesting question about the Green function for damped harmonic oscillators. I would like to clarify a few points about the Green function and its solutions for the damped harmonic oscillator.

Firstly, the Green function is a mathematical tool that helps us solve differential equations by breaking them down into simpler components. In the case of the undamped harmonic oscillator, the Green function is used to solve the equation with a delta function on the right-hand side. This is because the delta function represents an impulse or sudden force acting on the system at a specific time. By using the Green function, we can find the solution for any arbitrary force F(t).

Now, in the case of the damped harmonic oscillator, the Green function is still used to solve the equation with a delta function on the right-hand side. However, the solution will also include a transient part, as you correctly pointed out. This transient part represents the initial response of the system to the delta function before reaching its steady-state solution.

In the forum post you mentioned, the author is only considering the steady-state solution for the damped harmonic oscillator, which is why the Green function is written in the form G = A e^{\frac{{\omega(t-t')}}{2Q} + i\omega(t-t')} + complex conjugate. This is a valid solution for the steady state, but it does not capture the transient response of the system.

To find the complete solution, we need to consider both the steady-state and transient solutions. This can be done by writing the Green function as G = G_{ss} + G_{tr}, where G_{ss} is the steady-state solution and G_{tr} is the transient solution. By adding these two solutions, we can obtain the complete solution for the damped harmonic oscillator.

In summary, the Green function used for the damped harmonic oscillator only provides the steady-state solution. To find the complete solution, we need to consider both the steady-state and transient solutions. I hope this clarifies your doubts about the Green function for damped harmonic oscillators.