DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

Hamiltonian vector field and Hamiltonian function

Definition

For $H \in C^{\infty} (M)$ , its Hamiltonian vector field is the vector field $X_{H}$ determined by the library sign convention

$ι_{X_{H}} ω = d H .$

A vector field $X$ is Hamiltonian if $ι_{X} ω = d H$ for some smooth function $H$ ; such an $H$ is a Hamiltonian function for $X$ . A function and its vector field are distinct data, and completeness of $X_{H}$ is not assumed.

Depends on

Symplectic form and symplectic manifold

Used by

Motion of a completely integrable Hamiltonian is linear on invariant tori Corollary
Hamiltonian flows are complete on every symplectic manifold False statement
n independent first integrals automatically form a completely integrable system False statement
A Hamiltonian is conserved along its own flow Proposition
Hamiltonians for a fixed vector field differ by a locally constant function Proposition
The cotangent lift of a vector field is Hamiltonian Proposition
Hamilton equations in canonical cotangent coordinates Theorem
Hamiltonian vector fields exist uniquely for smooth functions Theorem
Symplectic vector fields modulo Hamiltonian vector fields are first de Rham cohomology Theorem

Dependency tree · two levels

2 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)