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.

Fibre derivative or Legendre map of a Lagrangian

Definition

For a smooth Lagrangian $L : T Q \to R$ , its fibre derivative or Legendre map is the fibre-preserving smooth map

$F L : T Q \to T^{*} Q, (F L (q, v)) (w) = {\frac{d}{d s} ∣}_{s = 0} L (q, v + s w) .$

In bundle coordinates it is $F L (q^{i}, v^{i}) = (q^{i}, p_{i})$ with $p_{i} = \partial L / \partial v^{i}$ . This coordinate formula also shows smoothness and that the base point $q$ is unchanged.

Used by

Regular and hyperregular Lagrangian Definition

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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