x = position, r = amplitude, ω = angular frequency, t = time

  A very important dynamical problem is that of mass attracted to a point by a force proportional to the distance from the point. In one dimension we write this as F = −kx, which on substituting F = ma becomes a = - k m x . We redefine k m as ω².

  Since acceleration is the second derivative of position with respect to time, we see that this equation is a second-order differential equation, written thus: ⅆ 2 x ⅆ t 2 + ω 2 x = 0 , the solution of which is x = r sin(ωt + f) , where f is an arbitrary constant. Setting f = 0 we recover the equation above.

  Any system governed by such a force displays oscillatory behaviour, the position and acceleration varying with a frequency ω. Such motion is called simple harmonic motion, and any body obeying the simple harmonic equation of motion is called a harmonic oscillator.