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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Constraint consistency for a regular first-order model

Physical scope

Conditional local result for a specified regular constraint surface and total-Hamiltonian ansatz; no universal termination or global quotient claim.

Statement

Let $I \subset R$ be an open interval and $(P, ω)$ a symplectic manifold. Let $χ_{a} : I \times P \to R$ be smooth and suppose the covectors $d_{z} χ_{a} (λ, z)$ are linearly independent at every $z \in N_{λ} : = {z : χ_{1} (λ, z) = \dots = χ_{r} (λ, z) = 0}$ ; this is the regular constraint condition of Regular primary constraints and total generator on each slice. Let $H_{c} : I \times P \to R$ be a smooth ambient extension of a specified descended Hamiltonian. Work in a Darboux chart with coordinates $(q^{i}, p_{i})$ and local potential $θ = p_{i} d q^{i}$ , so $d θ = - ω$ under the convention $ω = d q^{i} \land d p_{i}$ ; paths stay in this chart. For any compact interval $[a, b] \subset I$ with $a < b$ , take paths $z : [a, b] \to P$ that extend smoothly to a neighborhood of $[a, b]$ and stay in the chart, and multipliers $u^{a}$ that extend smoothly to a neighborhood of $[a, b]$ . Their action is

$I [z, u] = \int_{a}^{b} [θ (\dot{z}) - H_{c} (λ, z) - u^{a} (λ) χ_{a} (λ, z)] d λ .$

Use $C^{2}$ one-parameter variations of $z$ with fixed endpoints, whose paths remain in the chart, and unrestricted $C^{2}$ one-parameter variations of $u$ , all defined on a common neighborhood of $[a, b]$ . Assume $(z, u)$ is stationary: ${\frac{d}{d ϵ} ∣}_{ϵ = 0} I [z_{ϵ}, u_{ϵ}] = 0$ for every such variation through $(z, u)$ . These classes satisfy the path and fixed-endpoint hypotheses of Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression. Stationarity implies $z (λ) \in N_{λ}$ and $\dot{z} = X_{H_{c} (λ, \cdot)} + u^{a} (λ) X_{χ_{a} (λ, \cdot)}$ . Along this trajectory, preservation of the moving level set requires

$\partial_{λ} χ_{a} + {χ_{a}, H_{c} (λ, \cdot)} + u^{b} (λ) {χ_{a}, χ_{b}} = 0 at (λ, z (λ)) .$

For constraints independent of $λ$ , the first term is zero.

Facts & Assumptions

Given: An open interval, a symplectic $P$ , a smooth constraint family with regular slices $N_{λ}$ , a smooth ambient $H_{c}$ , and a smooth pair $(z, u)$ stationary for the displayed action under the stated variations.

[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 primary constraints and total generator: at each fixed parameter, the independent constraint differentials define the tangent space as the intersection of their kernels.

[F2]

Canonical Poisson brackets and local Hamiltonian flow: $ι_{X_{H}} ω = d H$ , $d F (X_{H}) = {F, H}$ , and the stated bracket convention applies.

[F3]

Darboux theorem: locally a symplectic form has canonical coordinates $ω = d q^{i} \land d p_{i}$ .

[F4]

Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression: fixed-endpoint first variation of a smooth first-order action vanishes exactly when its Euler–Lagrange equations hold.

Proof

1.1givenA1F3F4algebra

In a Darboux chart, the integrand is $p_{i} {\dot{q}}^{i} - H_{c} - u^{a} χ_{a}$ . Stationarity under independent variations of $u^{a}$ gives $χ_{a} = 0$ : apply [F4] to the combined coordinates $(q, p, u)$ , whose integrand has no $\dot{u}$ term, using its fixed-endpoint subclass. Stationarity under fixed-endpoint variations of $p_{i}, q^{i}$ gives ${\dot{q}}^{i} = \partial_{p_{i}} H_{T}$ and ${\dot{p}}_{i} = - \partial_{q^{i}} H_{T}$ , where $H_{T} = H_{c} + u^{a} χ_{a}$ . The equations hold on $[a, b]$ , including endpoints by continuity.

2.1givenF2step 1.1algebra

The multiplier functions depend on λ only, so $d_{z} H_{T} = d H_{c} + u^{a} d χ_{a}$ . By linearity of $ι_{X_{H}} ω = d H$ , $X_{H_{T}} = X_{H_{c}} + u^{a} X_{χ_{a}}$ . If instead phase-dependent extensions $u^{a} (z, λ)$ are used, the extra term $χ_{a} X_{u^{a}}$ vanishes on $N$ .

2.2givenF1step 1.1algebra

Differentiate $χ_{a} (λ, z (λ)) = 0$ along a solution; the chain rule yields $0 = \partial_{λ} χ_{a} + d_{z} χ_{a} (\dot{z})$ .

3.1givenF2step 2.1step 2.2algebra∎

Substitute the vector field from step 2.1 and use [F2]: $d_{z} χ_{a} (X_{H_{c}}) = {χ_{a}, H_{c}}$ and $d_{z} χ_{a} (X_{χ_{b}}) = {χ_{a}, χ_{b}}$ , giving the displayed consistency equation at each point of the trajectory. Requiring the total vector field to preserve every initial point of the moving constraint surface requires this equation throughout $N_{λ}$ . Its possible algebraic outcomes depend on its coefficient matrix and remaining constraints; no rank or termination premise was assumed.

Constraint consistency for a regular first-order model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources