Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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  • 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.
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A nondegenerate stationary envelope solves the Hamilton–Jacobi equation

Statement

If Sα(x0;α0)=0 and Sαα(x0;α0) is invertible, then locally Sα(x;α(x))=0 defines a C1 parameter α(x). The envelope u=S(x;α(x)) has Du=Sx(x;α(x)) and satisfies F(x,u,Du)=0.

Facts & Assumptions

Given: A C2 complete integral, a stationary point, and an invertible parameter Hessian there.

Proof

technique · direct
1.1

Apply the implicit function theorem to G(x,α)=Sα(x;α); its derivative in α is the invertible Sαα.

givenconstruct
2.1

The chain rule gives Du=Sx+SαDα=Sx on the stationary branch.

step 1.1givenalgebra
3.1

The complete-integral identity F(x,S,Sx)=0, evaluated at α(x) and using step 2.1, is F(x,u,Du)=0.

step 2.1given∎

Depends on

Used by

Dependency tree · two levels

12 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