Energy and Phase Portrait of the Oscillator

Conserved mechanical energy

For the undamped harmonic oscillator, the kinetic energy Ec = (1/2).m.v^2 and the elastic potential energy Ep = (1/2).k.x^2 continuously exchange with each other, but their sum Em = Ec + Ep remains constant:

Em = (1/2).k.A^2

This result is demonstrated by differentiating Em with respect to time and using the equation of motion: dEm/dt = 0.

Physical interpretation

When x is at its maximum (x = A), the speed is zero: all the energy is potential. When x = 0 (passing through equilibrium), the speed is at its maximum (v_max = A.omega0): all the energy is kinetic.

The phase portrait

By plotting v/omega0 as a function of x, you get a circle of radius A (an ellipse if you plot v as a function of x). This phase portrait is a powerful tool: each state of the system is a point, and its time evolution traces a closed trajectory for a conservative oscillator.

Position	Ec	Ep	v
x = A	0	max	0
x = 0	max	0	v_max
x = -A	0	max	0

Common pitfall

Watch out for homogeneity: Ep = (1/2).k.x^2 uses the displacement from equilibrium, not the absolute position if the equilibrium is not at the origin of the chosen frame. Always check where x = 0 is located before writing down the potential energy.