I was looking through one of the past Advanced Physics exam papers and I came across a question which is a little confusing at the end (the answer that is):
Consider a forced, damped simple harmonic oscillator, which obeys the equation ofmotion:
a) First consider the un-driven case ( )
Using the substitution
where A is an arbitrary constant, together with de Moivre’s theorem
derive the expression giving the damped SHM:
And the answer is too long so I'll just post the last few lines. Basically you let x(t)= Ae^(lambda)t dx/dt = A(lambda)e^(lambda)t and
d^2x/dt^2= A(lambda)^2 e^(lambda)t
sub d^2x/dt^2 into original equation (given above).
Divide by Ae^(lambda)t to elimintae on both sides.
Take everything over to the left hand side to get a quadratic in (lambda)^2.
x(t) = A e ^(-bt/2m) e^ (+-i (omega) t) where omega = the discriminant of the quadratic= i((k/m - b^2/4m^2)^1/2)
I apologise for the lack of LaTex but it would have taken like 30 minutes. The last 2 lines of the proof/answer are in LaTex:
(I know i have neglected the -iomega t case [as did the answers], but I believe this just gives )
Now here is where I get confused.
The next line is:
Why/how did they eliminate the sin? Is it because we are dealing with a real oscillator and an expression for oscillation cannot have an imaginary part? Does this hold with any example (that is if your dealing with something physical in the real world and you use complex numbers to prove, can you just eliminate the imaginary part at the end of the proof)?
Also if we end up eliminating the imaginary part in the end, why did we use complex numbers to prove in the first place? I am unsure why complex nmbers are used so rigorously to prove mathematical equations (especially if part of it is just deleted at the end of the proof).
Much thanks for any responses.