Integrable systems and Liouville tori

Let M have dimension 2n and let F=(f₁,…,f_n) be smooth functions with pairwise zero Poisson brackets. Assume their differentials are independent on a neighborhood of a compact connected regular common level L. The Hamiltonian fields X_{f_j} are then independent, tangent to L and commute. Their pairings vanish because the brackets do, so L is an n-dimensional Lagrangian submanifold. For an integrable Hamiltonian one can take H among the f_j or, more generally, H to be a function of their joint values near L.

Compactness makes each smooth tangent field complete on L. Their commuting flows define an Rⁿ action whose orbit is open because the fields span TL. Connectedness then makes this action transitive. The stabilizer is a discrete subgroup, and compactness of the quotient forces it to be a full lattice, so L is diffeomorphic to Rⁿ modulo a lattice, hence to a torus. Without compactness, the same argument can yield Euclidean directions instead of a full lattice. Without independence, the spanning and local-transitivity argument fails.

The Liouville–Arnold theorem strengthens the torus conclusion to a sufficiently small torus-fibered neighborhood of L with action–angle coordinates (ϑ,I), where each angle has period 2π and ω=Σdϑ_j∧dI_j. The integrals depend only on the actions there. Any saturation here refers to the restricted local fibration, not necessarily the full preimage of a base neighborhood including remote components. For H=h(I), Hamilton’s equations become Ī=0 and ϑ̇=∇h(I), so each invariant torus carries a constant-frequency linear flow. This neighborhood theorem is stated rather than fully proved here; obtaining a smoothly varying period lattice and canonical action normalization requires more than identifying the single torus topologically.

If a local primitive is available on the relevant neighborhood, actions can be normalized by integrating the canonical one-form around a chosen basis of cycles and dividing by 2π, up to permitted constants and integer changes of cycle basis. A global canonical primitive need not exist on the whole manifold, so this formula must be used with its local or cotangent-bundle setting stated. Resonance concerns frequencies on a particular torus: an integer relation among them changes the orbit closure but does not by itself destroy the independence of the original integrals or the surrounding action–angle coordinates.

A two-torus flow with nonzero frequencies is periodic when their ratio is rational; an irrational ratio gives a dense orbit on that torus. More generally the orbit closure is determined by integer relations among the frequencies. This is the linear-torus-flow theorem, not a statement that every bounded Hamiltonian trajectory is dense or periodic. A compact energy surface may contain nonintegrable dynamics, and enough conserved quantities without independence can fail to organize it into regular tori. The theorem’s structural tests must be checked before using its linear picture.

Singular levels can contain collapsed cycles, unstable equilibria and separatrices. For a pendulum, the separatrix is not a regular compact torus, and its limiting period diverges. Even when all nearby fibers are regular tori, global action coordinates can encounter topological obstructions such as monodromy. Local charts remain valuable without promising a single global labeling. These limitations are why an integrability claim should name its integrals, identify the regular domain and distinguish the neighborhood theorem from unresolved or singular regions of the system.