# [Physics] The meaning of “Angular frequency” of a spring that behaves as a simple harmonic oscilator

complex numbersharmonic-oscillatorspringwaves

In the simple harmonic oscillator model, we talk about the angular frequency and I understand that this tells something about the instantaneous angular velocity of the object. For example, a simple pendulum. But what does this mean when we apply this concept to an object attached to a spring that behaves as a simple harmonic oscillator? There is no instantaneous circular motion for the object, so how can we relate the angular frequency to a system like this?

The basic equation for the simple harmonic oscillator is $\frac{d^2x}{dt^2} =- \omega^2 x$. One solution that works here is $x=x_0 e^{i\omega t}$. Notice how the complex conjugate, $x^* = x_0^* e^{-i\omega t}$ is also a valid solution. The reason for this is because, while the body may be oscillating in one dimensional motion, it can be thought of as going either clockwise or counterclockwise in circle in the complex plane, where its angular velocity is either $i\omega$ for counterclockwise or $-i\omega$ for clockwise.