DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

Poisson bracket on a symplectic manifold

Definition

For $F, G \in C^{\infty} (M)$ , the Poisson bracket in the library convention is

${F, G} : = ω (X_{F}, X_{G}) = d F (X_{G}) = X_{G} (F) = - X_{F} (G) .$

All four formulas use $ι_{X_{H}} ω = d H$ . In particular, the order in the observable formula is important: evolution by $H$ differentiates $F$ as $X_{H} (F) = {F, H}$ .

Depends on

Hamiltonian vector fields exist uniquely for smooth functions

Used by

First integral and Poisson-commuting functions Definition
n independent first integrals automatically form a completely integrable system False statement
Coordinate formula for the Poisson bracket Proposition
Observable evolution equation Proposition
The Poisson bracket is bilinear, skew, and a derivation in each entry Proposition
The Hamiltonian vector-field map is a Lie antihomomorphism Theorem

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)