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.

First integral and Poisson-commuting functions

Definition

A smooth function $F$ is a first integral of a Hamiltonian $H$ if $F$ is constant along every integral curve of $X_{H}$ , on that curve's local maximal domain. Functions $F_{1}, \dots, F_{k}$ Poisson commute or are in involution if ${F_{i}, F_{j}} = 0$ for all $i, j$ . The first definition does not presume that the Hamiltonian flow is complete.

Depends on

Poisson bracket on a symplectic manifold

Used by

Completely integrable Hamiltonian system Definition
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)