LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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The Charpit momentum equation from differentiating Hamilton–Jacobi

Statement

If $u \in C^{2}$ solves $F (x, u, D u) = 0$ , put $P = D u (X)$ and let $\dot{X} = F_{p} (X, u (X), D u (X))$ . Then

$\dot{P} = - F_{x} - P F_{z},$

so the momentum equation is forced by differentiating the PDE.

Facts & Assumptions

Given: A $C^{2}$ classical solution and the stated projected characteristic.

Proof

technique · direct

1.1

Differentiate $F (x, u (x), D u (x)) = 0$ in $x$ to get $F_{x} + F_{z} D u + D^{2} u F_{p} = 0$ .

givenalgebra

1.2

Along $X$ , the chain rule gives $\dot{P} = D^{2} u (X) \dot{X} = D^{2} u (X) F_{p}$ .

givenalgebra

2.1

Combining the two identities and $P = D u (X)$ gives $\dot{P} = - F_{x} - P F_{z}$ .

step 1.1step 1.2algebra∎

Depends on

Used by

Quadratic Hamilton–Jacobi data produce an explicit caustic time Example

Dependency tree · two levels

10 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

Part I: Explicit methods — Lecture notes for MA342H (standard reference, not scraped)