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

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A symplectic non-Hamiltonian vector field on the two-torus

Example

On $T^{2} = R^{2} / Z^{2}$ with $ω = d x \land d y$ , the vector field $X = \partial_{x}$ is symplectic but is not Hamiltonian.

Facts & Assumptions

Given: The standard quotient coordinates, so $d x$ and $d y$ descend to global one-forms.

[F1]

A vector field is symplectic exactly when its contraction with $ω$ is closed. A vector field is symplectic iff $ι_{X} ω$ is closed.

[F2]

A symplectic field is Hamiltonian exactly when the cohomology class of its contraction with $ω$ vanishes. Symplectic vector fields modulo Hamiltonian vector fields are first de Rham cohomology.

Verification

technique · direct

1.1

Contraction gives $ι_{X} ω = d y$ , which is closed; hence [F1] shows that $X$ is symplectic.

F1givenalgebra

2.1

On the closed loop $γ (t) = [(0, t)]$ , $0 \leq t \leq 1$ , one has $\int_{γ} d y = 1$ . Every exact one-form has zero integral around a closed curve by the fundamental theorem of calculus, so $d y$ is not exact. Its class is nonzero, and [F2] proves that $X$ is not Hamiltonian.

F2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)