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I'm trying to numerically solve the differential equations of motion in a system with multiple springs of very high frequency. Because the solution is often a combination of rapidly-oscillating sine and cosine waves, it takes a very long time for a numerical solver to find the answer.

However, the fourier transform of the solution is probably fairly simple - it's nearly a delta function peaked at the frequency (or frequencies) at which the system oscillates.

Since Fourier transforms turn $\frac{df}{dx}$ into $2 \pi i \xi \hat{f}(\xi)$, it would seem that we could turn all of Newton's equations of the form $F_k=m\frac{d^2}{dt^2} x_k$ (representing, say, the $k$th particle or rigid body in a complicated system) into $\hat{F_k} = m (2 \pi i t)^2 \hat{x_k}$ (where the Fourier transform is taken with respect to time). Then, we could solve all of these simpler equations for the values of the $\hat{x_k}$, then take the Fourier transform (numerically).

So, do people ever apply Fourier transforms to the case of many-body systems with some harmonic oscillators?

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Correction to the question (v1): The $t$ appearing in the last equation $\hat{F_k} = m (2 \pi i t)^2 \hat{x_k}$ should be replaced with the dual variable (=frequency). – Qmechanic Jun 4 '11 at 17:43

Maybe they do; but Fourier transforms have an inherent flaw which makes them less than useful for such cases. That flaw is that they are transforms of steady state conditions: The initial and final conditions of the system are assumed to be the same; and transients are not considered.

The transform for which you seek is the Laplace transform. Laplace is a very close cousin to Fourier, but takes into account initial conditions and allows you to inspect transients and the final state of the system.

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