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

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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.

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Action–angle coordinates for the harmonic oscillator

Example

For the oscillator $H = (p^{2} + Ω^{2} q^{2}) / 2$ with $Ω > 0$ , remove the equilibrium. With a radian angle $ϕ \in R / (2 π Z)$ ,

$q = \sqrt{\frac{2 J}{Ω}} \sin ϕ, p = \sqrt{2 J Ω} \cos ϕ, J = \frac{H}{Ω} .$

Then $ω = d ϕ \land d J$ . In the page's period-one convention, $θ = ϕ / (2 π)$ and $I = 2 π J = 2 π H / Ω$ are action–angle coordinates.

Facts & Assumptions

Given: The standard form $d q \land d p$ and positive frequency $Ω$ .

[F1]

Period-one action–angle coordinates satisfy $ω = d θ \land d I$ . Action and angle coordinates.

Verification

technique · direct

1.1givenalgebra

Differentiating the displayed substitution gives $d q \land d p = d ϕ \land d J$ ; its Jacobian coefficient is $\cos^{2} ϕ + \sin^{2} ϕ = 1$ . Also direct substitution gives $H = Ω J$ .

2.1F1step 1.1∎

Since $d θ \land d I = (d ϕ / (2 π)) \land (2 π d J) = d ϕ \land d J$ , [F1] applies. Moreover $H = Ω I / (2 π)$ , so the library Hamiltonian convention gives $\dot{θ} = Ω / (2 π)$ , equivalently $\dot{ϕ} = Ω$ . Thus the displayed substitution explicitly supplies the action–angle coordinates. If angle has period $2 π$ instead, the corresponding action is $J = H / Ω$ ; the factor $2 π$ is purely normalization.

Depends on

Action and angle coordinates

Used by

Nothing in the library uses this result yet.

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)