Physic Labs

Analytical mechanics

Liouville’s theorem

Illustrate Liouville's theorem: Hamiltonian flow preserves phase-space volume (∇(q,p)⋅z˙=0\nabla_{(q,p)}\cdot\dot z = 0). Observe the orbit winding on a 2-dimensional invariant torus and vary the frequency ratio ω2/ω1\omega_2/\omega_1 to see the orbit structure change while the torus — a level set of the constants of motion — stays invariant.

Advanced

Equipment

  • 3D torus canvas with the orbit winding around it (drag to rotate, arrow keys change the view)
  • Slider “Tỉ lệ quấn ω₂/ω₁” (readout 0.xx–1.70)
  • “Tạm dừng / Chạy tiếp” button
  • Readout describing the torus as a 2-dimensional level set of an integrable system

Procedure

  1. Observe the orbit on the torus

    Watch the red curve wind around the torus surface in the canvas; drag or use the arrow keys to rotate the view. The readout notes that the torus illustrates a 2-dimensional level set of a two-degree-of-freedom integrable system — the surface J1=J_1 = const, J2=J_2 = const. Every initial point on this surface stays on it: the phase flow preserves both the set and its volume.

  2. Change the winding ratio ω₂/ω₁

    Move the “Tỉ lệ quấn ω₂/ω₁” slider and watch the readout beside it (shown as value/100). When the ratio is near simple rationals (0.50, 1.00, 1.50) the orbit nearly closes after a few turns; an irrational ratio makes the curve cover the torus densely. Press “Tạm dừng” to inspect the orbit shape, then “Chạy tiếp” to resume.

  3. Connect to phase-volume conservation

    Note that whatever the ω₂/ω₁ ratio, the curve stays confined to the same torus — it neither spills off nor collapses to a point. This pictures two results: torus invariance (because the actions JiJ_i are conserved) and Liouville's theorem preserving phase volume, ∇(q,p)⋅z˙=0\nabla_{(q,p)}\cdot\dot z = 0. Compare two far-apart ratio values and predict that at an irrational ratio the orbit will fill the torus.

Simulation

Experiment history

In 1838 the French mathematician Joseph Liouville proved that the phase flow generated by Hamilton's equations is incompressible: a region of initial states in phase space (q,p)(q, p) deforms but keeps its volume, expressed by ∇(q,p)⋅z˙=0\nabla_{(q,p)}\cdot\dot z = 0. The result later became the foundation of statistical mechanics — a probability density ρ\rho on phase space obeys a continuity equation. The geometric side of the simulation concerns orbit structure: for an integrable system with nn degrees of freedom, the action–angle theory of Delaunay (1840s), stated in full by Liouville (1855) and proved rigorously in the modern form by V. I. Arnold, shows that compact level sets are nn-dimensional tori on which motion is a linear flow with frequencies ωi\omega_i. On a 2-torus, the orbit closes when ω2/ω1\omega_2/\omega_1 is rational and covers the torus densely when irrational — exactly what the figure illustrates.

Related physicists

Isaac Newton

Related library topics

Liouville’s theorem