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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Hamiltonian vector fields exist uniquely for smooth functions

Statement

For every $H \in C^{\infty} (M)$ on a symplectic manifold $(M, ω)$ , there is a unique smooth vector field $X_{H}$ satisfying $ι_{X_{H}} ω = d H$ .

Facts & Assumptions

Given: A smooth function $H$ on $(M, ω)$ .

[F1]

The defining equation for $X_{H}$ is $ω^{♭} (X_{H}) = d H$ . Hamiltonian vector field and Hamiltonian function.

Proof

technique · direct

1.1

Nondegeneracy says that $ω^{♭} : T M \to T^{*} M$ is a fibrewise linear isomorphism, and its local matrix and inverse are smooth.

given

2.1

Thus $X_{H} = (ω^{♭})^{- 1} (d H)$ is smooth, satisfies [F1], and is the only possible solution because $ω^{♭}$ is injective.

F1step 1.1∎

Depends on

Hamiltonian vector field and Hamiltonian function

Used by

Poisson bracket on a symplectic manifold Definition
Time-dependent Hamiltonian vector field and flow Definition

Dependency tree · two levels

2 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)