PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Commuting Hamiltonian vector fields integrate to a local $R^{n}$ -action

Statement

On the regular locus of a completely integrable system, the fields $X_{F_{1}}, \dots, X_{F_{n}}$ integrate to a local $R^{n}$ -action. On a compact invariant regular fibre their restrictions are complete, so the action is global on that fibre.

Facts & Assumptions

Given: A completely integrable system and its Hamiltonian vector fields.

[F1]

Zero Poisson brackets make the local Hamiltonian flows commute. Hamiltonian flows commute iff their Hamiltonians Poisson commute up to locally constant bracket, Two vector fields commute if and only if their local flows commute.

[F2]

On a regular fibre the fields are tangent and span its tangent spaces. Regular common level sets are Lagrangian submanifolds.

Proof

technique · direct

1.1

Let $ϕ_{i}^{t}$ be the local flow of $X_{F_{i}}$ . Pairwise involution in complete integrability and [F1] make these flows commute. Therefore $(t_{1}, \dots, t_{n}) \cdot p = ϕ_{1}^{t_{1}} \circ \dots \circ ϕ_{n}^{t_{n}} (p)$ is independent of the order and satisfies the action law wherever both sides are defined.

F1given

1.2

By [F2], every regular fibre is invariant under all these flows. On a compact fibre, a maximal trajectory of any restricted smooth field cannot escape in finite time: a convergent subsequence near a finite endpoint and local ODE existence would extend it. Thus every restricted flow is complete.

F2given

2.1

Substituting the complete commuting restricted flows into the formula of step 1.1 defines a global $R^{n}$ -action on the compact fibre. Without compactness, only the local action is asserted.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)

Commuting Hamiltonian vector fields integrate to a local RnRn-action