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

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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.

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The cotangent lift of a vector field is Hamiltonian

Statement

Assume ${A C}_{ω}$ . Let $Y$ be a vector field on $Q$ and let $Y^{#}$ be the infinitesimal generator of the inverse-transpose cotangent lifts of its local flow. Then $Y^{#}$ is Hamiltonian for

$H_{Y} (q, p) = p (Y_{q}) .$

Facts & Assumptions

Given: The cotangent lift convention and the canonical form $ω_{c a n} = - d λ$ .

[F1]

Cotangent lifts preserve $λ$ and the canonical symplectic form. Cotangent lifts are symplectomorphisms.

[F2]

The library Hamiltonian equation is $ι_{X} ω = d H$ . Hamiltonian vector field and Hamiltonian function.

Proof

technique · direct

1.1

In coordinates $Y = Y^{i} (q) \partial_{q^{i}}$ , differentiation of the inverse-transpose lift gives $Y^{#} = Y^{i} \partial_{q^{i}} - p_{j} (\partial_{q^{i}} Y^{j}) \partial_{p_{i}}$ . Also $H_{Y} = p_{j} Y^{j}$ .

F1givenalgebra

2.1

Contracting with $ω_{c a n} = \sum_{i} d q^{i} \land d p_{i}$ gives $ι_{Y^{#}} ω_{c a n} = Y^{i} d p_{i} + p_{j} (\partial_{q^{i}} Y^{j}) d q^{i} = d (p_{j} Y^{j}) = d H_{Y}$ . By [F2], $Y^{#} = X_{H_{Y}}$ .

F2step 1.1algebra∎

Depends on

Used by

Angular momentum as a cotangent-lift Hamiltonian Example

Dependency tree · two levels

11 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)