Trajectory of a Harmonic Oscillator in Phase Space

You can observe how the trajectory of a harmonic oscillator in phase space evolves in time and how it depends on the characteristic values of the oscillator: the amplitude , the frequency , and the damping constant . In addition, the energy as a function of time is shown.

The phase space is a two-dimensional space spanned by the variables and , the displacement and momentum of the object. Because the simple harmonic motion is periodic, its trajectory is a closed curve, an ellipse. In this Demonstration, the axis is scaled so the ellipse is shown as a circle if the amplitude is set to its maximum value. The area of the ellipse is equal to the product of the energy and the cycle duration of the oscillator, so that in case of energy loss because of damping the ellipse converts to a logarithmic spiral.

The values shown are based on the values of the parameters in their SI units and the oscillator mass is 1 kg.