separatrix

A separatrix is a curve in phase space that divides regions of qualitatively different behaviour. Cross it, and the system does something fundamentally different — not just more or less of the same thing.

For a simple pendulum, the separatrix is the figure-eight curve in the (θ, ω) phase plane that passes through the unstable equilibrium at θ = ±π. Inside the loops, the pendulum oscillates back and forth — this is libration. Outside the loops, the pendulum has enough energy to swing over the top and keep rotating in one direction. On the separatrix itself, the pendulum takes infinite time to asymptotically approach the vertical, never quite reaching it.

The energy on the separatrix equals exactly mgl — the potential energy at the top. Any trajectory with less energy is trapped inside; any trajectory with more escapes into rotation. The separatrix is the watershed between two fundamentally different kinds of motion.