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.

Lagrangian action functional on curves

Definition

Let $Q$ be a smooth configuration manifold and let $L : T Q \to R$ be a smooth Lagrangian. For a $C^{1}$ curve $γ : [a, b] \to Q$ , its action is

$S_{L} (γ) = \int_{a}^{b} L (γ (t), \dot{γ} (t)) d t .$

For a variational problem, the endpoints are fixed: an admissible smooth variation $γ_{s}$ satisfies $γ_{s} (a) = γ (a)$ and $γ_{s} (b) = γ (b)$ . A $C^{2}$ curve is stationary if the derivative of the action at $s = 0$ vanishes for every such variation.

Used by

Euler–Lagrange equations Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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