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Hamiltonian Phase-Space Flow

What you are seeing: the orbits of Hamilton's equations in the (q,p)(q, p) plane. Click anywhere to release a tracer at (q0,p0)(q_0, p_0); the integrator steps in time and traces an orbit at constant energy H(q,p)H(q, p). For the pendulum, you can see librations (closed loops) below the separatrix E=1E = 1 and rotations (open curves) above.

Figure 1. Phase-space orbits and the flow field of a Hamiltonian.
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show flow

WHAT TO TRY

  • Click anywhere to release a tracer: it rides the Hamiltonian flow, looping on a closed orbit. For the pendulum, orbits inside the separatrix librate (swing) and orbits outside rotate (go over the top).
  • Switch the system between pendulum, harmonic oscillator and double well: the phase portrait reshapes, but every orbit stays on a constant-energy contour, since energy is conserved.
  • Seed tracers near the separatrix: they crawl, because the period diverges at the unstable fixed point. That slowing is the fingerprint of the saddle that divides the flow.