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

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The canonical Liouville vector field on a cotangent bundle is radial in momenta

Statement

Assume ${A C}_{ω}$ . For $ω_{c a n} = - d λ$ on $T^{*} Q$ , the Liouville vector field is

$Z = \sum_{i} p_{i} \frac{\partial}{\partial p_{i}} .$

Facts & Assumptions

Given: Canonical coordinates $(q^{i}, p_{i})$ on $T^{*} Q$ .

[F1]

$λ = \sum_{i} p_{i} d q^{i}$ and $ω_{c a n} = \sum_{i} d q^{i} \land d p_{i}$ . Tautological one-form on a cotangent bundle.

[F2]

The Liouville equation is $ι_{Z} ω = - λ$ . Liouville vector field on an exact symplectic manifold.

Proof

technique · direct

1.1F1algebra

For the displayed radial field, contraction with [F1] gives $ι_{Z} ω_{c a n} = - \sum_{i} p_{i} d q^{i} = - λ$ .

2.1F2step 1.1∎

Nondegeneracy makes the solution of [F2] unique, so this radial field is the canonical Liouville field. Its local flow is $(q, p) \mapsto (q, e^{t} p)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)