Liouville–Arnold theorem

Let $(M^{2 n}, ω)$ be a $2 n$ -dimensional symplectic manifold.
Let $F = (F_{1}, \dots, F_{n}) : M \to R^{n}$ be a smooth map whose components satisfy:

(Involution)

{F_{i}, F_{j}} = 0 for all 1 \leq i, j \leq n .

(Independence)
The differentials $d F_{1}, \dots, d F_{n}$ are linearly independent on a dense open subset of $M$
( $rank d F = n$ there).
Let $c \in R^{n}$ be a regular value of $F$ , and set

M_{c} := F^{- 1} (c) \subset M .

(i) If $M_{c}$ is compact and connected, then $M_{c}$ is diffeomorphic to an $n$ -dimensional torus $T^{n}$ .

(ii) There exist an open neighborhood $U$ of $M_{c}$ in $M$ and a smooth diffeomorphism

Φ : U \to T^{n} \times B (with B \subset R^{n} open)

such that:

In the coordinates $(Q_{1}, \dots, Q_{n}, P_{1}, \dots, P_{n})$ given by $Φ$ ,
$ω = \sum_{i = 1}^{n} d P_{i} \land d Q_{i}$
(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:
${\dot{P}}_{i} = 0, {\dot{Q}}_{i} = \frac{\partial H}{\partial P_{i}} =: ω_{i} (P) .$
Thus the flow is linear on each invariant torus:
$Q_{i} (t) = Q_{i} (0) + ω_{i} t (\mod 2 π), P_{i} (t) = const .$

The level sets $M_{c} = F^{- 1} (c)$ described in the theorem are precisely what are referred to as invariant Lagrangian tori. They are invariant because the Hamiltonian flow generated by any of the commuting functions $F_{i}$ is confined to the torus see Liouville foliation. They are Lagrangian because they are $n$ -dimensional submanifolds of the $2 n$ -dimensional symplectic manifold $(M, ω)$ on which the symplectic form $ω$ vanishes when restricted to the submanifold, i.e., $ω |_{M_{c}} = 0$ (see Lagrangian submanifold). This condition is satisfied because the level sets are locally defined by $d P_{i} = 0$ (since $P_{i}$ are constant on $M_{c}$ ), and thus $ω = \sum_{i = 1}^{n} d P_{i} \land d Q_{i}$ restricts to zero. The compactness and connectivity ensure the smooth topology of a torus $T^{n}$ .

Dynamics on the Torus: The dynamics is quasi-periodic in the sense of the linear evolution of the angle variables, given by $Q (t) = Q (0) + ω (P) t (\mod 2 π)$ . Here, the frequency vector $ω (P) = \nabla_{P} H$ determines the topology of the orbits. If the frequencies $ω_{1}, \dots, ω_{n}$ are incommensurable (linearly independent over the rationals $Q$ ), the trajectory never closes and is dense on the torus $T^{n}$ (ergodic motion). Conversely, if resonance relations exist (rational dependencies), the motion is periodic or confined to a lower-dimensional subtorus. Thus, the system behaves as a superposition of $n$ independent oscillators, returning arbitrarily close to its initial state without necessarily having a fixed period.

Remarks:

The $P_{i}$ are the action variables, constant on each torus.
The $Q_{i}$ are the angle variables, $2 π$ -periodic coordinates along the torus.
The theorem is local in $P$ but global in $Q$ on $T^{n}$ .
The assumption “compact and connected” for $M_{c}$ is crucial for the torus conclusion.