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We have just started learning small oscillations at grad level and our professor pulled out the technique of Fourier Analysis to illustrate its power by applying it to the case of a simple harmonic oscillator.

The differential equation is: $\ddot x+w_o^2x=0$

At this point he said, using the definition of fourier transform i.e. $$x(t)=\int_{-\infty}^{+\infty}\tilde x(w)\exp(iwt)\ dw$$

we get: $$(-w^2+w_0^2)\tilde x(w)=0$$ or $\tilde{x}(w)=A\delta(w-w_o)+B\delta(w+w_o)$

My problems:

  1. What does $\tilde x(w)$ mean physically?
  2. I have looked at other answers from the site and I can't understand the meaning of a negative $w$. It would be great if this doubt is cleared as well.

Any help is appreciated.

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  • $\begingroup$ Do you understand how a sine and a cosine (physics) are made up by positive and negative frequency Fourier components? $\endgroup$ Commented Sep 27, 2021 at 15:18
  • $\begingroup$ @CosmasZachos Do you mean $\cos wt= (e^{iwt}+e^{-iwt})/2$? $\endgroup$
    – Physiker
    Commented Sep 27, 2021 at 15:42
  • $\begingroup$ Of course. Negative frequencies provide needed mathematical components for physical ones. $\endgroup$ Commented Sep 27, 2021 at 15:44
  • $\begingroup$ So no other use? There is no physical sense in them apart from elegance? $\endgroup$
    – Physiker
    Commented Sep 27, 2021 at 17:48
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    $\begingroup$ Probably yes, in your misbegotten question. Mathematical consistency involving complex numbers is not mere elegance, in the long run. $\endgroup$ Commented Sep 27, 2021 at 17:54

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When Fourier transform is used in the form with complex exponential, the Fourier coefficients satisfy $$ [\tilde{x}(\omega)]^*=\tilde{x}(-\omega), $$ which follows from the fact that $x(t)$ is a real function, i.e. $x(t)^*=x(t)$. Thus, the negative frequencies here do not really play independent role. One could write the Fourier transform as an expansion in sines and cosines, considering only positive frequencies, but it would be cumbersome, see here.

In a general setting $\tilde{x}(\omega)$ is the (complex) amplitude of the oscillations, i.e., it contains the information about their real amplitude and their phase. However, in the case of the oscillator this interpretation is only confusing, since there is only one frequency and the delta functions appear.

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