The simple harmonic oscillator is perhaps the most fundamental model in all of physics. It describes any system where a restoring force is directly proportional to the displacement from equilibrium.

Hooke's Law and Dynamics

The classical example is a mass attached to a spring, governed by Hooke's Law: $F = -kx$. When combined with Newton's second law, this yields the differential equation $\ddot{x} + \omega^2 x = 0$, where $\omega = \sqrt{k/m}$ is the natural angular frequency. The solutions to this equation are pure sinusoidal waves, demonstrating endless, periodic exchange between kinetic and potential energy.