Angle vs. Angular Velocity — the geometry of motion
Energy (Amplitude)
1.50
Damping
0.020
Speed
1.5x
The phase portrait reveals the pendulum's full dynamics. Closed loops = oscillation (below separatrix).
Wavy curves = full rotation (above separatrix). The separatrix (shown in gold) is the boundary —
an infinite-time trajectory ending at the unstable equilibrium (top). Damping spirals all orbits toward rest.