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

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A vector field is symplectic iff $ι_{X} ω$ is closed

Statement

A vector field $X$ on $(M, ω)$ is symplectic if and only if the one-form $ι_{X} ω$ is closed.

Facts & Assumptions

Given: A vector field $X$ on a symplectic manifold $(M, ω)$ .

[F1]

Symplectic means $L_{X} ω = 0$ . Symplectic vector field.

[F2]

Cartan's formula is $L_{X} ω = d (ι_{X} ω) + ι_{X} d ω$ . Cartan's magic formula.

Proof

technique · direct

1.1

Since $d ω = 0$ , [F2] reduces to $L_{X} ω = d (ι_{X} ω)$ .

F2given

2.1

Therefore the left side vanishes exactly when $ι_{X} ω$ is closed, which is precisely the equivalence in [F1].

F1step 1.1∎

Depends on

Used by

Dependency tree · two levels

9 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)

A vector field is symplectic iff ιXωιX​ω is closed