Dynamical systems and ergodic theory Hamiltonian dynamics Theorem

Liouville–Arnold theorem

Builds on Symplectic manifold Poisson bracket Hamiltonian vector field Lagrangian submanifold Torus Integrable system Action-angle coordinates

Statement

Let $(M, \omega)$ be a symplectic manifold of dimension $2n$ . Let $F_1, \ldots, F_n \colon M \to \R$ be smooth functions with Poisson brackets $\{F_i, F_j\} = 0$ for all $i, j$ , and let $H = F_1$ . Let $F = (F_1, \ldots, F_n) \colon M \to \R^n$ , let $c \in \R^n$ be a regular value of $F$ , and let $M_c = F^{-1}(c)$ . Then:

$M_c$ is a smooth Lagrangian submanifold of dimension $n$ . It is invariant under the flows of the Hamiltonian vector fields $X_{F_1}, \ldots, X_{F_n}$ .
Each compact connected component of $M_c$ is diffeomorphic to the torus $\mathbb{T}^n = \R^n / 2\pi \Z^n$ .
On such a component there are angle coordinates $\theta \in \mathbb{T}^n$ in which the flow of $X_H$ is linear: $\theta(t) = \theta(0) + t \, \nu(c)$ for a vector $\nu(c) \in \R^n$ .
A neighbourhood of such a component has action-angle coordinates $(I, \theta)$ in which $H$ is a function of $I$ only.

Remarks

Since the $F_i$ Poisson-commute, the flows of $X_{F_1}, \ldots, X_{F_n}$ commute and preserve $M_c$ . They give an action of $\R^n$ on each component of $M_c$ , which is locally free because the $dF_i$ are independent there. A compact connected manifold with a transitive locally free $\R^n$ -action is $\R^n / \Lambda$ for a lattice $\Lambda$ , that is, a torus. The actions are built from the integrals $\oint p \, dq$ over the basic cycles of the tori.

Example: for two uncoupled harmonic oscillators on $\R^4$ , with $F_1 = \frac{1}{2}(p_1^2 + q_1^2) + \frac{1}{2}(p_2^2 + q_2^2)$ and $F_2 = \frac{1}{2}(p_2^2 + q_2^2)$ , the level sets with $c_1 > c_2 > 0$ are products of two circles, that is, $2$ -tori.

Compactness is necessary for the torus conclusion. For a free particle on $\R$ with $H = p^2/2$ , the level set $H = c > 0$ is two lines, not circles, and the motion is not periodic.

Going deeper

For a student

The theorem is the structure theorem for integrable systems. It turns the algebraic condition of commuting integrals into a geometric picture: a fibration of phase space by invariant Lagrangian tori, with linear flow on each torus. The frequency vector $\nu$ decides the type of motion. When its entries are rationally independent, each orbit is dense in its torus. When they satisfy an integer relation, the torus is filled by invariant tori of lower dimension, and no orbit is dense in it. This picture is the starting point of KAM theory, which asks which tori survive a small perturbation of $H$ .

Reading path

To understand this page from the ground up, read these 70 pages in order. The foundations come first.

Action-angle coordinates concept
Element definition
Hamiltonian vector field concept
Integrable system concept
Lagrangian submanifold concept
Poisson bracket concept
Proposition concept
Real numbers object
Symplectic manifold concept
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Complement definition
Function definition
Basis of a topology definition
Subspace topology definition
Equivalence relation definition
Product topology definition
Preimage of a set definition
Metric space definition
Codomain definition
Function composition definition
Injective function definition
Equivalence class definition
Open ball definition
Surjective function definition
Metric topology definition
Quotient set definition
Bijective function definition
Open set definition
Quotient topology definition
Inverse function definition
Hausdorff space definition
Open cover definition
Continuous map of topological spaces definition
Closed set definition
Compact space definition
Complement of an open set is closed statement
Homeomorphism definition
Universal property of the quotient topology statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example
Torus example
Liouville–Arnold theorem statement

Tag	`PNQCW85HTJ`
URI	`m2://PNQCW85HTJ`
Kind	statement, theorem
Writing	W0 stub
MSC	37J35, 70H06, 53D12
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About	Integrable system, Action-angle coordinates
Uses	Symplectic manifold, Poisson bracket, Hamiltonian vector field, Lagrangian submanifold, Torus, Integrable system, Action-angle coordinates
Same as	Wikidata Q14943261 Wikipedia Liouville–Arnold theorem