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.

A Hamiltonian is conserved along its own flow

Statement

$H$ is constant along every integral curve of $X_{H}$ .

Facts & Assumptions

Given: A Hamiltonian $H$ and an integral curve $γ$ of $X_{H}$ .

[F1]

The convention is $d H = ι_{X_{H}} ω$ . Hamiltonian vector field and Hamiltonian function.

Proof

technique · direct

1.1F1given

Along $γ$ , $\frac{d}{d t} H (γ (t)) = d H (X_{H}) = ω (X_{H}, X_{H}) = 0$ by alternation.

2.1step 1.1∎

Hence $H \circ γ$ is constant on every connected time interval in the maximal domain. This proves conservation without assuming completeness.

Depends on

Hamiltonian vector field and Hamiltonian function

Used by

The finite modified Hamiltonian has a bounded one-step defect Lemma

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)