ExampleConstruction: AI-adaptedVerification: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Legendre transform of a natural mechanical Lagrangian

Example

For a supplied Riemannian metric $g$ and smooth potential $V$ ,

$L (q, v) = \frac{1}{2} g_{q} (v, v) - V (q)$

has $p = g_{q}^{♭} (v)$ , inverse velocity $v = g_{q}^{♯} (p)$ , and Hamiltonian $H (q, p) = \frac{1}{2} g_{q}^{- 1} (p, p) + V (q)$ .

Facts & Assumptions

Given: The displayed natural Lagrangian.

[F1]

The general natural-mechanical calculation gives hyperregularity, the Legendre map, and the kinetic-plus-potential Hamiltonian. A natural mechanical Lagrangian gives the kinetic-plus-potential Hamiltonian.

Verification

technique · direct

1.1F1givenalgebra

Differentiating $L (q, v + s w)$ at $s = 0$ gives $g_{q} (v, w)$ , so its fibre derivative is $p = g_{q}^{♭} (v)$ . Positive definiteness makes $g_{q}^{♭}$ invertible, with inverse $g_{q}^{♯}$ , and [F1] gives hyperregularity.

2.1F1step 1.1algebra∎

The energy is $p (v) - L = g (v, v) - \frac{1}{2} g (v, v) + V = \frac{1}{2} g (v, v) + V$ . Substituting $v = g^{♯} p$ gives $H (q, p) = \frac{1}{2} g^{- 1} (p, p) + V (q)$ , as asserted.

Depends on

A natural mechanical Lagrangian gives the kinetic-plus-potential Hamiltonian

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

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