Liouville–Arnold theorem

Let $(M^{2n},\omega )$ be a $2n$-dimensional symplectic manifold.
Let $F=(F_1,\dots ,F_n):M\to {\mathbb{R}}^n$ be a smooth map whose components satisfy:

1. (Involution)

2. (Independence)
   The differentials $d{F}_1,\dots ,d{F}_n$ are linearly independent on a dense open subset of $M$
   ($\mathrm{rank}dF=n$ there).
   Let $c\in {\mathbb{R}}^n$ be a regular value of $F$, and set

(i) If ${\mathcal{M}}_c$ is compact and connected, then ${\mathcal{M}}_c$ is diffeomorphic to an $n$-dimensional torus ${\mathbb{T}}^n$.

(ii) There exist an open neighborhood $U$ of ${\mathcal{M}}_c$ in $M$ and a smooth diffeomorphism

such that:

• In the coordinates $(Q_1,\dots ,Q_n,P_1,\dots ,P_n)$ given by $\mathrm{\Phi }$,

  $$\omega =\sum _{i=1}^{n}d{P}_i\wedge d{Q}_i$$

  (canonical Darboux form). They are called action-angle variables.

• The functions $F_i$ depend only on $P$, i.e. $F_i=f_i(P)$,

• The Hamiltonian vector field $X_H$ (if $H$ is one of the $F_i$ or any smooth function of them) satisfies:

  $${\dot{P}}_i=0,{\dot{Q}}_i=\frac{\partial H}{\partial P_i}=:{\omega }_i(P).$$

  Thus the flow is linear on each invariant torus:

  $$Q_i(t)=Q_i(0)+{\omega }_{i}t(\mathrm{mod}2\pi ),P_i(t)=\text{const}.$$

Remarks:

• The $P_i$ are the action variables, constant on each torus.
• The $Q_i$ are the angle variables, $2\pi$-periodic coordinates along the torus.
• The theorem is local in $P$ but global in $Q$ on ${\mathbb{T}}^n$.
• The assumption “compact and connected” for ${\mathcal{M}}_c$ is crucial for the torus conclusion.