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.

Action and angle coordinates

Definition

Action–angle coordinates on a neighbourhood $U$ fibred by Lagrangian tori are a diffeomorphism

$(I, θ) : U ⟶ B \times (R / Z)^{n},$

where $B \subseteq R^{n}$ is open, such that the components $I = (I_{1}, \dots, I_{n})$ and $θ = (θ_{1}, \dots, θ_{n})$ have fibres $I = c o n s t a n t$ and

$ω = \sum_{i = 1}^{n} d θ_{i} \land d I_{i} .$

This page uses period one for every angle. Replacing angles by period $2 π$ rescales the corresponding actions and formulas. The order $d θ_{i} \land d I_{i}$ is forced by the library convention $ι_{X_{H}} ω = d H$ : it gives ${\dot{θ}}_{i} = \partial H / \partial I_{i}$ .

Depends on

Symplectic form and symplectic manifold

Used by

Action–angle coordinates for the harmonic oscillator Example
Liouville–Arnold action–angle theorem 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

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