Harmonic oscillator

Example

Let $m > 0$ and $k > 0$ be constants, and let $\omega_0 = \sqrt{k/m}$. The equation of the harmonic oscillator is $$m\, x''(t) + k\, x(t) = 0, \quad \text{that is} \quad x'' + \omega_0^2\, x = 0.$$ It models a mass $m$ on a spring with stiffness $k$, where $x$ is the displacement from the rest position. Its characteristic roots are $\pm i \omega_0$, so its solutions are $x(t) = c_1 \cos(\omega_0 t) + c_2 \sin(\omega_0 t) = A \cos(\omega_0 t + \varphi)$ with constants $A \ge 0$ and $\varphi$.