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 $\varphi$.

Remarks

Each nonzero solution is periodic with period $2 \pi / \omega_0$. The period does not depend on the amplitude $A$. The energy $E = \frac{1}{2} m (x')^2 + \frac{1}{2} k x^2$ is constant along each solution. It is a first integral. With friction, the equation becomes the damped harmonic oscillator.

Explanations

L1 human

The spring pulls the mass back with a force proportional to the displacement. The mass overshoots the rest position, and the motion repeats forever as a sine wave. A stiffer spring or a lighter mass gives a faster oscillation.