Action-angle variables

Action-angle variables are canonical coordinates for integrable Hamiltonian systems, pairing conserved actions with angles that advance uniformly in time.

Hamiltonian mechanics: A formulation of classical mechanics in which a Hamiltonian function governs the evolution of positions and momenta. Action-angle coordinates are canonical coordinates within this broader framework.
Action integral: A phase-space integral of momentum around a closed orbit, conventionally divided by 2π. Each action is computed by integrating the conjugate momentum around a periodic cycle.
Perturbation theory: A collection of methods for approximating solutions when a system differs slightly from a solvable one. Action-angle variables organize small departures from integrable motion.
Action-angle coordinates for the harmonic oscillator: A canonical representation of the harmonic oscillator in which the action is constant and the angle increases linearly. This exactly solvable case shows the simplest form of the transformation.
Symplectic geometry: The geometry of manifolds equipped with a closed, nondegenerate differential two-form. Canonical transformations preserve the symplectic structure that action-angle coordinates must retain.
Angle variable: A periodic canonical coordinate that specifies position along a cycle and typically advances uniformly in time. It tracks phase along the orbit paired with each conserved action.
KAM theory: A theory describing which invariant tori persist under small perturbations of integrable Hamiltonian systems. Its persistence results are naturally expressed in action-angle coordinates.
Pendulum: A mechanical system consisting of a mass suspended from a pivot and moving under gravity. Its librations admit action-angle coordinates, while the separatrix exposes their limits.
Liouville–Arnold theorem: A theorem describing the invariant tori and linear motion of regular, integrable Hamiltonian systems. It establishes when invariant tori admit action-angle coordinates.
Liouville integrability: The condition that a Hamiltonian system has enough independent, mutually commuting conserved quantities for integration by quadratures. Action-angle coordinates generally require this level of integrability.

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