# nLab Liouville integrable system

## Surveys, textbooks and lecture notes

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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 sens 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 $(X, \omega)$.

Revised on October 15, 2012 20:46:54 by Zoran Škoda (161.53.130.104)