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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Hamiltonians for a fixed vector field differ by a locally constant function

Statement

If $H$ and $K$ are Hamiltonian functions for the same vector field on $M$ , then $H - K$ is locally constant, hence constant on each connected component. Conversely, adding a locally constant function does not change the Hamiltonian vector field.

Facts & Assumptions

Given: Smooth functions $H, K$ and the Hamiltonian convention.

[F1]

A Hamiltonian for $X$ satisfies $d H = ι_{X} ω$ . Hamiltonian vector field and Hamiltonian function.

Proof

technique · direct

1.1

If both functions generate $X$ , [F1] gives $d (H - K) = 0$ . In a connected coordinate ball, integration along line segments shows that a smooth function with zero differential is constant; hence $H - K$ is locally constant and therefore constant on each connected component.

F1given

2.1

Conversely, if $c$ is locally constant then $d c = 0$ , so $d (H + c) = d H$ and [F1] gives $X_{H + c} = X_{H}$ .

F1step 1.1∎

Depends on

Hamiltonian vector field and Hamiltonian function

Used by

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)