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.

Energy and Hamiltonian of a hyperregular Lagrangian

Definition

The energy of a Lagrangian $L$ is

$E_{L} (q, v) = F L (q, v) (v) - L (q, v) .$

If $L$ is hyperregular, its associated Hamiltonian on $T^{*} Q$ is

$H = E_{L} \circ (F L)^{- 1} .$

Equivalently, if $p = \partial L / \partial v$ and $v = v (q, p)$ is the smooth inverse Legendre relation, then $H (q, p) = p_{i} v^{i} - L (q, v)$ . Hyperregularity is what makes this a globally defined smooth function.

Depends on

Regular and hyperregular Lagrangian

Used by

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)