What is Phase portrait, fixed points, separatrix? — Physics

Definition 2.6 University Physics — Year 3 · Chapter 2 — Hamiltonian Mechanics

The phase portrait of a system is the family of its trajectories in phase space. A fixed point is a state where both canonical equations vanish — an equilibrium: a minimum of the potential appears as a centre surrounded by closed curves, a maximum as a saddle through which passes a separatrix, the trajectory that divides phase space into regions of qualitatively different motion.

The pendulum’s phase portrait: level curves of H. Closed ovals (swings) circulate clockwise around the centre; above and below the separatrix, the pendulum rotates. The saddles at ± 180 are the inverted equilibrium. — The pendulum’s phase portrait: level curves of $H$ . Closed ovals (swings) circulate clockwise around the centre; above and below the separatrix, the pendulum rotates. The saddles at $\pm 180^\circ$ are the inverted equilibrium.

Examples

Example 2.7 (The pendulum’s phase portrait)

For $H = p^2/2m\ell^2 - mg\ell\cos\theta$ : closed ovals around $(0, 0)$ — librations, the ordinary swings; wavy curves at $|p|$ large — rotations, the pendulum turning over the top; and through the saddles at $\theta = \pm\pi$ the separatrix, of energy $E = +mg\ell$ , on which the pendulum takes an infinite time to reach the top. A state on the separatrix is the swing launched exactly hard enough to arrive at the inverted position with nothing to spare.

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Examples

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