Liouville–Arnold theorem

The Liouville–Arnold theorem states that compact, connected, regular common level sets of a completely integrable Hamiltonian system are invariant tori. Locally, the motion on each torus is linear in action–angle coordinates.

Hamiltonian system: A dynamical system whose evolution is generated by a Hamiltonian function on phase space. The theorem describes the motion of systems governed by Hamiltonian dynamics.
Symplectic manifold: A smooth manifold equipped with a closed, nondegenerate differential two-form. Hamiltonian systems and their canonical coordinates are defined on symplectic manifolds.
Action variable: An integral quantity associated with a cycle of an integrable system, used as a canonical coordinate. Action variables label invariant tori and remain constant during the motion.
Joseph Liouville: A French mathematician whose work included the theory of integrable mechanical systems. His name marks the integrability condition at the theorem's foundation.
Poisson commutativity: A condition in which the Poisson bracket of two functions vanishes. Commuting integrals are the conserved quantities required for complete integrability.
First integral: A function on a dynamical system's state space that remains constant along trajectories. The theorem uses conserved functions to define its common level sets.
Angle variable: A periodic canonical coordinate that tracks position along an invariant torus. Angle variables increase linearly with time on the torus.
Siméon Denis Poisson: A French mathematician who developed the bracket now used to express commutation of integrals. Poisson brackets provide the language for the theorem's commuting first integrals.
Regular value theorem: A theorem stating that the preimage of a regular value under a smooth map is a submanifold. Regular common levels have the smooth structure needed to become invariant tori.
Complete integrability: The existence of sufficiently many independent, mutually commuting first integrals for a system. Complete integrability supplies the theorem's family of conserved quantities.

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