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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Observable evolution equation

Statement

If $γ$ is an integral curve of $X_{H}$ , then every observable $F$ satisfies

$\frac{d}{d t} F (γ (t)) = {F, H} (γ (t)) .$

Facts & Assumptions

Given: Smooth functions $F, H$ and an integral curve $γ$ of $X_{H}$ .

[F1]

${F, H} = X_{H} (F)$ . Poisson bracket on a symplectic manifold.

Proof

technique · direct

1.1

The chain rule and $\dot{γ} = X_{H} \circ γ$ give $\frac{d}{d t} (F \circ γ) = d F (X_{H}) \circ γ$ .

given

2.1

By [F1], $d F (X_{H}) = X_{H} (F) = {F, H}$ , proving the formula on the entire local domain of the curve.

F1step 1.1∎

Depends on

Poisson bracket on a symplectic manifold

Used by

F is a first integral of H iff F and H Poisson commute Proposition

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)