ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

The constrained pendulum multiplier follows from tangency

Example

For a planar pendulum of mass $m > 0$ and rod length $ℓ > 0$ , let $M = m I_{2}$ , $V (q) = m g q_{2}$ , and $c (q) = (∥ q ∥_{2}^{2} - ℓ^{2}) / 2$ . If $c (q) = 0$ and $q^{T} p = 0$ , the multiplier in the constrained equations is $λ (q, p) = \frac{∥ p ∥_{2}^{2} / m - m g q_{2}}{ℓ^{2}} .$ At the bottom rest state $q = (0, - ℓ)$ , $p = 0$ , this gives $λ = m g / ℓ$ and zero acceleration.

Facts & Assumptions

Given: $M = m I_{2}$ , the constraint and potential in the Example, and consistent data $c (q) = 0$ , $q^{T} p = 0$ .

[F1]

Regular holonomic phase space and constrained dynamics: $A (q) = J (q) M^{- 1} J (q)^{T}$ and $λ = A^{- 1} (D^{2} c [v, v] - J M^{- 1} \nabla V)$ , where $v = M^{- 1} p$ .

[F2]

The constrained Hamiltonian vector field has a unique local flow: these equations define the unique local constrained flow for consistent data in the regular chart.

Verification

1.1givenF1algebra

Here $J (q) = q^{T}$ , $A (q) = q^{T} q / m = ℓ^{2} / m$ , $D^{2} c [v, v] = ∥ p ∥_{2}^{2} / m^{2}$ , and $J M^{- 1} \nabla V = g q_{2}$ . Substitution into [F1] gives $λ = (∥ p ∥_{2}^{2} / m - m g q_{2}) / ℓ^{2}$ .

2.1F1F2step 1.1algebra∎

At $q = (0, - ℓ)$ and $p = 0$ , the multiplier is $m g / ℓ$ . The acceleration is $\dot{p} = - m g (0, 1) - λ (0, - ℓ) = (0, 0)$ ; moreover $c (q) = 0$ and $q^{T} p = 0$ , so this is a consistent equilibrium of the local flow [F2].

Depends on

Used by

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Dependency tree · two levels

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Sources

Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric Numerical Integration, 2nd ed., Chapter VII.1.1, p. 237 (standard reference, not scraped)
Robert I. McLachlan, Klas Modin, Olivier Verdier, and Matt Wilkins, Geometric Generalisations of SHAKE and RATTLE, Foundations of Computational Mathematics 14 (2014), 339–370, §5.1 (standard reference, not scraped)