TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Darboux theorem

Statement

Assume ${A C}_{ω}$ . For every point $p$ of a $2 n$ -dimensional symplectic manifold $(M, ω)$ , there are coordinates $(q^{1}, \dots, q^{n}, p_{1}, \dots, p_{n})$ centred at $p$ in which

$ω = \sum_{i = 1}^{n} d q^{i} \land d p_{i} .$

Facts & Assumptions

Given: ${A C}_{ω}$ , a symplectic manifold $(M, ω)$ , and $p \in M$ .

[F1]

A nondegenerate alternating form has a symplectic basis. Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even.

[F2]

Symplectic forms that agree as tensors along a closed embedded submanifold and have a locally symplectic interpolation are related by a local symplectomorphism fixed there. Relative Moser theorem.

Proof

technique · direct

1.1

By [F1], choose a chart $x = (q^{'}, p^{'})$ centred at $p$ whose differential identifies $ω_{p}$ with $ω_{s t d} = \sum_{i} d q^{' i} \land d p_{i}^{'}$ at the origin. The forms $ω$ and $x^{*} ω_{s t d}$ agree at $p$ . Their convex interpolation is nondegenerate on a neighbourhood of $p$ for every $t \in [0, 1]$ , after shrinking once, because nondegeneracy is open and the parameter interval is compact.

F1given

2.1

Apply [F2] to the closed submanifold ${p}$ . It gives a local diffeomorphism $ϕ$ fixing $p$ with $ϕ^{*} (x^{*} ω_{s t d}) = ω$ . Therefore the components of $x \circ ϕ$ are the required coordinates. When $n = 0$ , the empty coordinate list already works.

F2step 1.1∎

Depends on

Used by

Symplectic manifolds have no local invariants beyond dimension Corollary
Darboux coordinates for a nonconstant area form Example
Darboux theorem makes all symplectic manifolds globally symplectomorphic False statement

Dependency tree · two levels

11 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)