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.

Time-dependent Hamiltonian vector field and flow

Definition

Assume ${A C}_{ω}$ , let $(M, ω)$ be a symplectic manifold, and let $I \subseteq R$ be an interval. A smooth function $H : I \times M \to R$ determines the time-dependent Hamiltonian vector field $X_{H_{t}}$ by

$ι_{X_{H_{t}}} ω = d (H_{t}) .$

Its Hamiltonian evolution is the two-time local evolution $Φ_{t, s}$ satisfying $\partial_{t} Φ_{t, s} = X_{H_{t}} \circ Φ_{t, s}$ and $Φ_{s, s} = id$ , wherever it exists. Neither this definition nor pointwise existence of $X_{H_{t}}$ asserts completeness of the evolution.

Depends on

Used by

Time-dependent Hamiltonian evolution is symplectic Proposition

Dependency tree · two levels

10 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

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)