CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

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Motion of a completely integrable Hamiltonian is linear on invariant tori

Statement

In action–angle coordinates, if $H = h (I)$ then

${\dot{I}}_{i} = 0, {\dot{θ}}_{i} = \frac{\partial h}{\partial I_{i}}, θ (t) = θ (0) + t \nabla h (I (0)) (m o d Z^{n}) .$

Facts & Assumptions

Given: Action–angle coordinates near an invariant Liouville torus and a Hamiltonian $H = h (I)$ .

[F1]

The convention $ι_{X_{H}} ω = d H$ defines the Hamiltonian vector field. Hamiltonian vector field and Hamiltonian function.

Proof

technique · direct

1.1

The supplied equality $H = h (I)$ says that $H$ has no $θ$ dependence. Write $X_{H} = \sum_{i} (a_{i} \partial_{θ_{i}} + b_{i} \partial_{I_{i}})$ . Since $ω = \sum_{i} d θ_{i} \land d I_{i}$ , contraction gives $ι_{X_{H}} ω = \sum_{i} (a_{i} d I_{i} - b_{i} d θ_{i})$ . Comparing this with $d H = \sum_{i} h_{I_{i}} d I_{i}$ by [F1] yields $a_{i} = h_{I_{i}}$ and $b_{i} = 0$ .

F1givenalgebra

2.1

The action values are constant, so the vector $\nabla h (I (0))$ is constant along the orbit. Integrating on $R^{n} / Z^{n}$ gives the displayed linear motion.

step 1.1∎

Depends on

Hamiltonian vector field and Hamiltonian function

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

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