DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Liouville vector field on an exact symplectic manifold

Definition

Let $(M, ω)$ be exact with a specified primitive $ω = - d λ$ . The Liouville vector field associated with $λ$ is the unique vector field $Z$ satisfying

$ι_{Z} ω = - λ .$

Cartan's formula gives $L_{Z} ω = d (- λ) + ι_{Z} d ω = - d λ = ω$ . Thus its local flow expands the symplectic form. The field depends on the chosen primitive $λ$ .

Depends on

Used by

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

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)