ExampleConstruction: AI-adaptedVerification: 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.

Angular momentum as a cotangent-lift Hamiltonian

Example

Assume ${A C}_{ω}$ . Fix $a \in R^{3}$ . The infinitesimal rotation $Y_{a} (q) = a \times q$ lifts to the Hamiltonian vector field on $T^{*} R^{3}$ with Hamiltonian

$H_{a} (q, p) = a \cdot (q \times p) .$

Facts & Assumptions

Given: Euclidean dot and cross products identify covectors with vectors.

[F1]

The cotangent lift of $Y$ has Hamiltonian $H_{Y} (q, p) = p (Y_{q})$ under the library convention. The cotangent lift of a vector field is Hamiltonian.

Verification

technique · direct

1.1

Insert $Y_{a} (q) = a \times q$ into [F1]. The scalar triple-product identity gives $H_{Y_{a}} (q, p) = p \cdot (a \times q) = a \cdot (q \times p)$ .

F1givenalgebra

2.1

Hence the infinitesimal cotangent-lifted rotation is $X_{H_{a}}$ . Varying $a$ shows that the vector-valued observable is $q \times p$ , the usual angular momentum.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)