# Liouville's Theorem Applied to 1D Harmonic Oscillator

Liouville's Theorem Applied to 1D Harmonic Oscillator

Time evolution generated by the action of a Hamiltonian preserves area in position-momentum phase space. Watch how an area deforms with time, eventually cycling back to its initial geometry. Using phase-space plots for the one-dimensional harmonic oscillator, two trajectories, each based on a specific initial value of a pair of momentum and position values, are plotted with controls to vary these phase-space points. A circular region approximately bounded by these trajectories is plotted between these points, then replotted at a few subsequent times. Controls are provided to vary the time of each of the time-evolved regions.