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.

Regular and hyperregular Lagrangian

Definition

A Lagrangian $L : T Q \to R$ is regular if its fibre Hessian

$(\frac{\partial^{2} L}{\partial v^{i} \partial v^{j}})$

is nonsingular at every point. Equivalently, its Legendre map $F L$ is a local diffeomorphism.

It is hyperregular if $F L : T Q \to T^{*} Q$ is a global fibre-preserving diffeomorphism. Hyperregularity implies regularity; local invertibility alone does not imply global bijectivity.

Depends on

Fibre derivative or Legendre map of a Lagrangian

Used by

Energy and Hamiltonian of a hyperregular Lagrangian Definition

Dependency tree · one level

1 result within one dependency step 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)