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

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

Statement

$F$ is a first integral of $H$ if and only if ${F, H} = 0$ on $M$ .

Facts & Assumptions

Given: Smooth functions $F, H$ on a symplectic manifold.

[F1]

Along every integral curve of $X_{H}$ , $\frac{d}{d t} F = {F, H}$ . Observable evolution equation.

[F2]

A first integral is constant on every such local curve. First integral and Poisson-commuting functions.

Proof

technique · direct

1.1

If ${F, H} = 0$ , [F1] makes the derivative of $F$ along every integral curve zero. Ordinary one-variable calculus makes $F$ constant on each interval domain, so it is a first integral by [F2].

F1F2givenalgebra

2.1

Conversely, through every $p \in M$ there is a local integral curve. If $F$ is a first integral, its derivative at time zero is zero; [F1] identifies it with ${F, H} (p)$ . Since $p$ was arbitrary, the bracket vanishes everywhere.

F1F2given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)

FF is a first integral of HH iff FF and HH Poisson commute