nLab Liouville integrable system

Idea

What is called Liouville integrability is one formalization of the notion of classical integrable system in physics.

A physical system is called Liouville integrable if it admits canonical coordinates and canonical momenta which are generated from the flow of a maximal set of commuting Hamiltonians.

Definition

A physical system given by a phase space symplectic manifold $(X, \omega)$ and equipped with a Hamiltonian $H_0 \in C^\infty(C)$ (generating time evolution) is said to be Liouville integrable or to be an integrable system in the sense of Liouville if

• there are $(dim X -1)$ other Hamiltonian functions $\{H_i\}_{i = 0}^{dim X -1}$

• such that

  1. these all commute under the Poisson bracket with each other;

  2. the flow of the corresponding Hamiltonian vector fields generates a polarization of $(X, \omega)$.