Hamiltonian phase space

Phase-space structure: Poincaré’s surface of section for the restricted three-body problem, KAM tori surviving and breaking in the standard map, torus breakdown in the non-integrable Henon-Heiles system, Liouville’s theorem via a sheared swarm of pendulums, action-angle variables for the simple pendulum, and Hamilton’s unified (q, p) canonical phase space itself, shared by a librating and a rotating pendulum.

The Henon-Heiles system: Poincare sections and KAM torus breakdown

The Henon-Heiles system: Poincare sections and KAM torus breakdown

Liouville’s theorem: a swarm of pendulums

Liouville's theorem: a swarm of pendulums

Action-angle variables for the pendulum

Action-angle variables for the pendulum

Hamilton’s canonical (q, p) phase space

Hamilton's canonical (q, p) phase space

Poincaré and the three-body problem

Poincaré and the three-body problem

The KAM theorem: which invariant tori survive a perturbation

The KAM theorem: which invariant tori survive a perturbation