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

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Hamiltonian flows preserve the symplectic form

Statement

Wherever the local flow $ϕ_{t}$ of a Hamiltonian vector field $X_{H}$ is defined, it preserves the symplectic form: $ϕ_{t}^{*} ω = ω$ . No completeness assertion is made.

Facts & Assumptions

Given: A Hamiltonian vector field and its local flow.

[F1]

A Hamiltonian field is symplectic, so $L_{X_{H}} ω = 0$ . A vector field is symplectic iff $ι_{X} ω$ is closed.

[F2]

A tensor is invariant under a local flow exactly when its Lie derivative along the generator vanishes. A tensor field is flow-invariant exactly when its Lie derivative vanishes.

Proof

technique · direct

1.1

Since $ι_{X_{H}} ω = d H$ is closed, [F1] gives $L_{X_{H}} ω = 0$ .

F1given

2.1

Apply [F2] on the domain of the local flow to obtain $ϕ_{t}^{*} ω = ω$ . Neither step extends the flow beyond its maximal domain.

F2step 1.1∎

Depends on

Used by

Hamiltonian flows are complete on every symplectic manifold False statement
Liouville volume preservation Theorem

Dependency tree · two levels

6 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)