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.

Coordinate formula for the Poisson bracket

Statement

Assume ${A C}_{ω}$ . In canonical cotangent coordinates,

${F, G} = \sum_{i} (\frac{\partial F}{\partial q^{i}} \frac{\partial G}{\partial p_{i}} - \frac{\partial F}{\partial p_{i}} \frac{\partial G}{\partial q^{i}}) .$

Facts & Assumptions

Given: Smooth functions $F, G$ in a canonical cotangent chart.

[F1]

Hamilton's equations give $X_{G} = \sum_{i} (G_{p_{i}} \partial_{q^{i}} - G_{q^{i}} \partial_{p_{i}})$ . Hamilton equations in canonical cotangent coordinates.

[F2]

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

Proof

technique · direct

1.1

Apply the vector field in [F1] to $F$ : $X_{G} (F) = \sum_{i} (G_{p_{i}} F_{q^{i}} - G_{q^{i}} F_{p_{i}})$ .

F1given

2.1

By [F2] this is ${F, G}$ , and commuting scalar factors gives the displayed formula with the library sign.

F2step 1.1algebra∎

Depends on

Used by

Poisson brackets in canonical coordinates Example

Dependency tree · two levels

11 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)