Separatrix

A separatrix is an invariant trajectory or surface that separates regions with qualitatively different motion. In a one-degree-of-freedom conservative system, it commonly lies on the energy level through an unstable equilibrium and separates oscillation from escape or rotation.

For the nondimensional simple pendulum $\ddot{x}=-\sin x$, the separatrix has energy ${\dot{x}}^2/2+1-\cos x=2$. Starting at $x=0$ with velocity $2$, its time to angle $a<\pi$ is $\log (\sec (a/2)+\tan (a/2))$. The saddle at angle $\pi$ is approached only as time tends to infinity.

// Target: classical-mechanics.bigb