Liouville–Arnold theorem

Statement

Preliminary definitions

Let ${\displaystyle (M^{2n},\omega )}$ be a ${\displaystyle 2n}$-dimensional symplectic manifold with symplectic structure ${\displaystyle \omega }$.

An integrable system on ${\displaystyle M^{2n}}$ is a set of ${\displaystyle n}$ functions on ${\displaystyle M^{2n}}$, labelled ${\displaystyle F=(F_1,\cdots ,F_n)}$, satisfying

• (Generic) linear independence: ${\displaystyle d{F}_1\wedge \cdots \wedge d{F}_n\ne 0}$ on a dense set
• Mutually Poisson commuting: the Poisson bracket ${\displaystyle (F_i,F_j)}$ vanishes for any pair of values ${\displaystyle i,j}$.

The Poisson bracket is the Lie bracket of vector fields of the Hamiltonian vector field corresponding to each ${\displaystyle F_i}$. In full, if ${\displaystyle X_H}$ is the Hamiltonian vector field corresponding to a smooth function ${\displaystyle H:M^{2n}\to \mathbb{R}}$, then for two smooth functions ${\displaystyle F,G}$, the Poisson bracket is ${\displaystyle (F,G)=[X_F,X_G]}$.

A point ${\displaystyle p}$ is a regular point if ${\displaystyle d{f}_1\wedge \cdots \wedge d{f}_n(p)\ne 0}$.

The integrable system defines a function ${\displaystyle F:M^{2n}\to {\mathbb{R}}^n}$. Denote by ${\displaystyle L_{\mathbf{c}}}$ the level set of the functions ${\displaystyle F_i}$, $${\displaystyle L_{\mathbf{c}}=\{x:F_i(x)=c_i\},}$$ or alternatively, ${\displaystyle L_{\mathbf{c}}=F^{-1}(\mathbf{c})}$.

Now if ${\displaystyle M^{2n}}$ is given the additional structure of a distinguished function ${\displaystyle H}$, the Hamiltonian system ${\displaystyle (M^{2n},\omega ,H)}$ is integrable if ${\displaystyle H}$ can be completed to an integrable system, that is, there exists an integrable system ${\displaystyle F=(F_1=H,F_2,\cdots ,F_n)}$.

Theorem

If ${\displaystyle (M^{2n},\omega ,F)}$ is an integrable Hamiltonian system, and ${\displaystyle p}$ is a regular point, the theorem characterizes the level set ${\displaystyle L_c}$ of the image of the regular point ${\displaystyle c=F(p)}$:

• ${\displaystyle L_c}$ is a smooth manifold which is invariant under the Hamiltonian flow induced by ${\displaystyle H=F_1}$ (and therefore under Hamiltonian flow induced by any element of the integrable system).
• If ${\displaystyle L_c}$ is furthermore compact and connected, it is diffeomorphic to the N-torus ${\displaystyle T^n}$.
• There exist (local) coordinates on ${\displaystyle L_c}$ ${\displaystyle ({\theta }_1,\cdots ,{\theta }_n,{\omega }_1,\cdots ,{\omega }_n)}$ such that the ${\displaystyle {\omega }_i}$ are constant on the level set while ${\displaystyle {\dot{\theta }}_i:=(H,{\theta }_i)={\omega }_i}$. These coordinates are called action-angle coordinates.

Examples of Liouville-integrable systems

A Hamiltonian system which is integrable is referred to as 'integrable in the Liouville sense' or 'Liouville-integrable'. Famous examples are given in this section.

Some notation is standard in the literature. When the symplectic manifold under consideration is ${\displaystyle {\mathbb{R}}^{2n}}$, its coordinates are often written ${\displaystyle (q_1,\cdots ,q_n,p_1,\cdots ,p_n)}$ and the canonical symplectic form is ${\displaystyle \omega =\sum _{i}d{q}_i\wedge d{p}_i}$. Unless otherwise stated, these are assumed for this section.

• Harmonic oscillator: ${\displaystyle ({\mathbb{R}}^{2n},\omega ,H)}$ with ${\displaystyle H(\mathbf{q},\mathbf{p})=\sum _i\left(\frac{p_i^2}{2m}+\frac{1}{2}m{\omega }_i^2{q}_i^2\right)}$. Defining ${\displaystyle H_i=\frac{p_i^2}{2m}+\frac{1}{2}m{\omega }_i^2{q}_i^2}$, the integrable system is ${\displaystyle (H,H_1,\cdots ,H_{n-1})}$.

• Central force system: ${\displaystyle ({\mathbb{R}}^6,\omega ,H)}$ with ${\displaystyle H(\mathbf{q},\mathbf{p})=\frac{{\mathbf{p}}^2}{2m}-U({\mathbf{q}}^2)}$ with ${\displaystyle U}$ some potential function. Defining the angular momentum ${\displaystyle \mathbf{L}=\mathbf{p}\times \mathbf{q}}$, the integrable system is ${\displaystyle (H,{\mathbf{L}}^2,L_3)}$.

• Integrable tops: The Lagrange, Euler and Kovalevskaya tops are integrable in the Liouville sense.