Simple harmonic motion

Remarks

Equivalently, $x$ is a nonzero solution of $\ddot x + \omega^2 x = 0$ for some $\omega > 0$, the equation of the harmonic oscillator. For a mass $m > 0$ on a spring of stiffness $k > 0$, the frequency is $\omega = \sqrt{k/m}$.

Example: $x(t) = 3 \sin(2t) = 3 \cos(2t - \pi/2)$ is simple harmonic motion with $A = 3$, $\omega = 2$ and period $\pi$. Check: $\ddot x = -12 \sin(2t) = -4 x$.

Non-example: $x(t) = e^{-t} \cos t$ solves $\ddot x + 2 \dot x + 2 x = 0$, but its amplitude decays, so it is not of the form $A \cos(\omega t + \varphi)$ with constant $A$; it is a damped oscillation.