TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Energy descends through an almost-regular Legendre map

Physical scope

Conditional theorem under the explicit almost-regular package; not a universal theorem for singular Lagrangians.

Statement

Let $L (t, q, v)$ be smooth on an open velocity domain and let $F L (t, q, v) = (t, q, L_{v})$ be almost regular in the sense that it is a surjective submersion onto a specified embedded image $N$ and every fibre is connected. Define $E_{L} = v^{i} L_{v^{i}} - L$ . Then there is a unique smooth function $h : N \to R$ with $E_{L} = h \circ F L$ . The assertion is local in the stated domains; no global inverse velocity map is asserted.

Facts & Assumptions

Given: Smooth $L$ , its fibre derivative, the specified embedded image $N$ , surjective-submersion property, and connected fibres.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Assume countable choice ACω, meaning every sequence of nonempty sets has a choice function; no full Axiom of Choice is assumed.

[F1]

Regular, hyperregular and almost-regular Legendre domains: all these almost-regular hypotheses are assumptions, not consequences of degeneracy.

[F2]

Fibre derivative or Legendre map of a Lagrangian: $p_{i} = L_{v^{i}}$ , with velocity derivative $L_{v^{i} v^{j}}$ .

[F3]

Local normal form for submersions: a submersion has smooth local sections through every point of its target.

Proof

1.1givenA1F2algebra

A tangent vector to a fibre of $F L$ has the form $(0, 0, δ v)$ with $L_{v^{i} v^{j}} δ v^{j} = 0$ , since the $(t, q)$ components of $F L$ are identity components and its $p$ -variation vanishes. Differentiating $E_{L}$ in this direction gives $d E_{L} (δ v) = v^{i} L_{v^{i} v^{j}} δ v^{j} = 0$ ; the $L_{v^{j}} δ v^{j}$ terms cancel.

2.1givenF1step 1.1algebra

On a connected fibre, a smooth function whose differential vanishes is constant: it is locally constant in coordinate neighborhoods, and a locally constant function on a connected space is constant. Hence $E_{L}$ is constant on each fibre of $F L$ . Define $h (y)$ to be this common value for any point above $y \in N$ ; surjectivity gives existence, and constancy gives well-definedness and uniqueness.

3.1givenF3step 2.1∎

For each $y \in N$ , [F3] gives a smooth local section $s$ of $F L$ . On that neighborhood $h = E_{L} \circ s$ , so $h$ is smooth. These local formulas agree on overlaps because the value is already uniquely constant on each fibre. Thus globally on $N$ , $E_{L} = h \circ F L$ .

Depends on

Dependency tree · 2 levels

Sources

Henneaux, Teitelboim and Regge, Constrained Hamiltonian Systems, Part 1