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Liouvillian flow on the pendulum phase space

Drop a Gaussian blob of 256 tracers anywhere in the pendulum phase space $(\theta, p)$. Each tracer evolves under the Hamiltonian $H = p^2/2 - \cos(\theta)$ via velocity-Verlet. The cloud rotates, shears, and filaments, but its phase-space area is preserved (Liouville's theorem). The dashed curve is the separatrix $p = \pm 2\cos(\theta/2)$: below it orbits librate, above it they rotate. Drag the dashed circle (handle) to set the blob center.

Figure 1. Pendulum phase space (theta, p) with a Gaussian tracer cloud evolving under H = p^2/2 − cos(theta). Method: 256 independent tracers integrated by velocity-Verlet from shared/js/engine/symplectic.js, with the symplectic Jacobian preserving the phase-space covariance determinant to within engine accuracy.

WHAT TO TRY

  • Vary each control and watch the rail readouts respond.
  • Compare the diagnostic plot against the live scene.