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: ${\displaystyle df_{1}\wedge \cdots \wedge df_{n}\neq 0}$ 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 ${\displaystyle H:M^{2n}\rightarrow \mathbb {R} }$, 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 ${\displaystyle df_{1}\wedge \cdots \wedge df_{n}(p)\neq 0}$.

The integrable system defines a function ${\displaystyle F:M^{2n}\rightarrow \mathbb {R} ^{n}}$. 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.