Double Pendulum Phase Portrait and Energy Conservation

Two bobs on rigid massless rods, both swinging under gravity, with the lower bob pinned to the end of the upper rod. The Lagrangian gives coupled velocity-dependent equations of motion; the qualitative behavior ranges from quasi-periodic at low energy to fully chaotic above the lowest saddle in the potential. Drag a bob to set its initial angle; double-click the canvas to zero out velocities.

Figure 1. Planar double pendulum integrated with velocity-Verlet plus a predictor-corrector for the qdot-dependent Christoffel terms (physics dt = 1 ms). Method: `shared/js/engine/symplectic.js`.

m₁ 1.0 kg

m₂ 1.0 kg

l₁ 1.00 m

l₂ 1.00 m

release 166°

pendulums 18

WHAT TO TRY

Watch the coloured spray: a thousandth of a radian apart at the start, the pendulums fan out completely within seconds and trace wildly different paths. That is chaos, sensitive dependence on the start.
The lower plot is the separation between them on a log axis. The straight climb is exponential blow-up; the dashed line is the fit and its slope is the Lyapunov exponent (the readout shows how often the separation doubles).
Add more pendulums for a denser, more beautiful fan, and use the release slider to set a gentle low angle: the motion turns regular, the spray collapses to one curve, and the divergence plot reports no blow-up. Chaos only switches on once you raise the arms.