Classical Motion and Phase Space for a Harmonic Oscillator

This Demonstration illustrates the classical harmonic motion of a particle governed by the Hamiltonian , where the scaled variables are defined as , . Here and are obtained by solving Hamilton's equations of motion, subject to the initial conditions and . The three panels animate synchronously: (1) the motion of the particle in the potential; (2) the phase space trajectory; and (3) the time series of and . In the upper two panels, the points with minimum (turning points) and maximum momentum are labeled with blue and green 's, respectively.