LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

The regular holonomic phase space is a symplectic cotangent bundle

Statement

Under the hypotheses of Regular holonomic phase space and constrained dynamics, let $Q_{r e g} = {q \in Q : rank J (q) = m}$ . It is a smooth manifold of dimension $d - m$ . Choose a local configuration chart $ϕ : O \subseteq R^{d - m} \to Q_{r e g}$ and write $T = D ϕ$ , $G = T^{T} M T$ . The cotangent coordinates $π = T^{T} p$ identify the part of $M$ over this chart locally with $T^{*} Q_{r e g}$ : the unique consistent ambient momentum representing $π$ is $p = M T G^{- 1} π$ . In these coordinates the restricted canonical one-form $p^{T} d q$ is $π^{T} d x$ , so the restricted two-form is $ω_{M} = \sum_{j = 1}^{d - m} d x_{j} \land d π_{j}$ . It is closed and nondegenerate, hence is symplectic on the regular phase manifold.

Facts & Assumptions

Given: The objects and regularity hypotheses in Regular holonomic phase space and constrained dynamics, and the canonical symplectic convention in Symplectic form and symplectic manifold.

[F1]

Regular holonomic phase space and constrained dynamics: $Q_{r e g} = {q \in Q : rank J (q) = m}$ , $M$ is constant SPD, and $M$ consists of $(q, p)$ with $q \in Q_{r e g}$ and $J M^{- 1} p = 0$ .

[F2]

The parametrized implicit function theorem with $C^{k}$ regularity: if a square block derivative of a $C^{k}$ map is invertible at a zero, its zero set is locally the graph of a unique $C^{k}$ function; this applies for every finite $k$ when the map is smooth.

[F3]

Symplectic form and symplectic manifold: a symplectic form is a smooth closed two-form that is pointwise nondegenerate.

[F4]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a composite is the composite of its derivatives.

Proof

1.1F1F2construct

At any $q \in Q_{r e g}$ , full row rank of $J (q)$ gives a permutation of coordinates $q = (x, y)$ for which the $m \times m$ block $D_{y} c (q)$ is invertible. Apply [F2] to the smooth map $c$ for each finite $k \geq 1$ . The resulting local graphs agree with the unique $C^{1}$ graph on overlaps, so that graph is $C^{k}$ locally for every $k$ and hence smooth. Thus locally $Q_{r e g}$ is $y = ψ (x)$ ; $ϕ (x) = (x, ψ (x))$ parametrizes it, and $D ϕ$ has rank $d - m$ with image $\ker J$ .

2.1step 1.1F1algebra

Define $G (x) = D ϕ (x)^{T} M D ϕ (x)$ . For every nonzero $ξ$ , $D ϕ ξ \neq 0$ and $ξ^{T} G (x) ξ = (D ϕ ξ)^{T} M (D ϕ ξ) > 0$ , so $G$ is invertible. Its entries are smooth, and $G^{- 1} = adj (G) / \det G$ is smooth because $\det G > 0$ . For a cotangent coordinate $π \in R^{d - m}$ , put $p = M D ϕ G^{- 1} π$ . Then $J M^{- 1} p = J D ϕ G^{- 1} π = 0$ and $D ϕ^{T} p = G G^{- 1} π = π$ . Conversely, if $J M^{- 1} p = 0$ , then $M^{- 1} p = D ϕ ξ$ for a unique $ξ$ , and $D ϕ^{T} p = G ξ$ ; hence the displayed formula gives the unique ambient $p$ representing $π$ . Both this map and its inverse $(q, p) \mapsto (x, D ϕ (x)^{T} p)$ are smooth, so $(x, π) \mapsto (ϕ (x), p)$ is a local diffeomorphism from $T^{*} Q_{r e g}$ onto the part of $M$ over this chart.

3.1F3F4step 2.1algebra

Let $θ_{0} = p^{T} d q$ be the ambient canonical one-form. Under the chart just constructed, $d q = D ϕ d x$ and $D ϕ^{T} p = π$ , so its pullback is $π^{T} d x$ . Therefore the pullback of $ω_{0} = - d θ_{0}$ is $\sum_{j = 1}^{d - m} d x_{j} \land d π_{j}$ , which is closed and nondegenerate.

4.1F3step 3.1∎

Every point of $M$ has a chart of this form; the local calculation proves the restricted two-form is closed and nondegenerate everywhere. By [F3] it is symplectic, as claimed.

Depends on

Used by

Dependency tree · two levels

15 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

Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric Numerical Integration, 2nd ed., Chapter VII.1, §§VII.1.1–VII.1.4 (standard reference, not scraped)
Robert I. McLachlan, Klas Modin, Olivier Verdier, and Matt Wilkins, Geometric Generalisations of SHAKE and RATTLE, Foundations of Computational Mathematics 14 (2014), 339–370 (standard reference, not scraped)