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.

Completely integrable Hamiltonian system

Definition

On a $2 n$ -dimensional symplectic manifold, a Hamiltonian system is completely integrable if it has smooth functions $F_{1} = H, F_{2}, \dots, F_{n}$ such that

${F_{i}, F_{j}} = 0$ for every $i, j$ ; and
$d F_{1}, \dots, d F_{n}$ are linearly independent on a dense open subset.

The map $F = (F_{1}, \dots, F_{n}) : M \to R^{n}$ is the integral map. The set on which $d F$ has rank $n$ is its regular locus; it is open and, by the preceding condition, dense. Both involution and independence are essential. Results about regular fibres apply only at regular values or specified regular components.

Depends on

First integral and Poisson-commuting functions

Used by

A singular common level need not be a torus Counterexample
Poisson-commuting functions with dependent differentials do not give Liouville–Arnold coordinates Counterexample
n independent first integrals automatically form a completely integrable system False statement
Commuting Hamiltonian vector fields integrate to a local ℝⁿ-action Proposition
Regular common level sets are Lagrangian submanifolds Proposition
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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)
Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)