The solution of the system of linear differential equations of the form
$$ \vec{x}''(t) =A\vec{x}(t)$$
Is given in the form:
$$ \vec{x}(t)=\sum_{j=1}^n \left( a_j \cos(\omega t) + b_j \sin (\omega t) \right) \vec{v} $$
The fourier series is given by:
$$ s_n = \frac{a_0}{2} + \sum_{n=1}^N \left(a_n \cos\left(\tfrac{2\pi}{P} nx \right) + b_n \sin\left(\tfrac{2\pi}{P} nx \right) \right) $$
This looks very similar. Is there a simple connection between these two?