The Simple Harmonic Oscillator

In general a particle in motion is subject to many forces. The harmonic oscillator, with potential equation
$V=x^2$
, gives rise to the equation
$\frac{d^2x}{dt^2}+ \omega^2 x=A$
has the motion
$x=\frac{A}{\omega^2}+B cos \omega t + C sin \omega t$
.
The simple harmonic oscillator is a useful model because as oscillation tends to zero, most motion becomes simple harmonic.
The vibrations of atoms in solids, the motions of pendulums, the heights of tides can all be modelled as simple harmonic, and the model can be modified to model many more systems - car damping systems, resonance and many others.