ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck pass

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.

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The free particle of the model: the flow $Q = q + (t / m) p$ generated by the first-kind function $S_{t} (q, Q) = - m ∣ Q - q ∣^{2} / (2 t)$ , the projection check and the exact change of $λ$

Physical scope

A worked generating-function computation for the declared one-label free-particle presentation: the explicit free Hamiltonian flow, its first-kind generating function $S_{t}$ with the mixed-Hessian determinant, the invertibility of the $(q, Q)$ -projection for $t \neq 0$ , the degeneration of the generating data at $t = 0$ , and the exact change of the canonical one-form along the flow. All quantities are model values of the declared idealization; no measurement is reported.

Example

Declare the one-label model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N = 1$ , mass $m > 0$ , time window $R$ , configuration domain $Q = R^{3}$ , zero potential $U \equiv 0$ and no residual forces (the free particle instance, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). By Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian the branch Hamiltonian is $H (x, p) = ∣ p ∣^{2} / (2 m)$ with Hamiltonian system $\dot{x} = m^{- 1} p$ , $\dot{p} = 0$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential). Then:

for every initial datum the maximal solution is

$Φ_{t} (x, p) = (x + (t / m) p, p) (t \in R),$

the free-particle flow of the presentation; 2. for $t \neq 0$ the function $S_{t} (q, Q) : = - \frac{m}{2 t} ∣ Q - q ∣^{2}$ is of class $C^{\infty}$ on $R^{3} \times R^{3}$ with $[S_{t}] = [m] [Q - q]^{2} / [t] = μ ℓ^{2} τ^{- 1}$ (action), $S_{t, q Q} = (m / t) I_{3}$ invertible everywhere, and the first-kind map it generates is exactly $Φ_{t}$ : indeed $p = S_{t, q} = m (Q - q) / t$ gives $Q = q + (t / m) p$ , and $P = - S_{t, Q} = m (Q - q) / t = p$ ; consequently $Φ_{t}$ is a canonical map of the presentation with $∣ \det D Φ_{t} ∣ = 1$ (Generating functions for a potential presentation: first- and second-kind data generate canonical maps locally, the local converse under an invertible projection, and the time-dependent Hamiltonian $K = H + \partial_{t} S$ clauses 1, Generating functions of the first and second kind for a local canonical map, The local map generated by a first- or second-kind generating function is a canonical $C^{2}$ local diffeomorphism); 3. projection check: the map $(q, p) \mapsto (q, Q) = (q, q + (t / m) p)$ has invertible derivative $(\begin{array}{cc} I_{3} & 0 \\ I_{3} & (t / m) I_{3} \end{array})$ for $t \neq 0$ , while the family degenerates at $t = 0$ : $S_{t}$ has no value there and the projection becomes singular, so this family does not represent the identity map; 4. the canonical one-form changes by an exact form along the flow: $Φ_{t}^{*} λ = λ + t d K$ with $K = ∣ p ∣^{2} / (2 m)$ , because $Φ_{t}^{*} λ = P \cdot d Q = p \cdot (d q + (t / m) d p) = λ + (t / m) d (∣ p ∣^{2} / 2)$ ; in particular $Φ_{t}^{*} d λ = d λ$ , recovering canonicity directly (The canonical one-form and the two-form of a potential presentation: $λ = p \cdot d x$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, A $C^{1}$ map between phase domains preserves the standard two-form iff $D C^{T} J D C = J$ , and a canonical diffeomorphism carries $X_{H \circ C}$ to $X_{H}$ ).

All quantities are model values of the declared idealization; no measurement is reported.

Facts & Assumptions

Given: The one-label free-particle instance with its Hamiltonian, flow and two-form, the generating datum $S_{t}$ , and the canonical one-form $λ = \sum_{a} p_{a} d q^{a}$ of the presentation.

[F1]

Model and Hamiltonian (Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model): $H = ∣ p ∣^{2} / (2 m)$ on $R^{6}$ with Hamilton system $\dot{x} = m^{- 1} p$ , $\dot{p} = 0$ , whose solutions are $x (t) = x_{0} + (t / m) p_{0}$ , $p (t) = p_{0}$ .

[F2]

Generating functions (Generating functions for a potential presentation: first- and second-kind data generate canonical maps locally, the local converse under an invertible projection, and the time-dependent Hamiltonian $K = H + \partial_{t} S$ , Generating functions of the first and second kind for a local canonical map, The local map generated by a first- or second-kind generating function is a canonical $C^{2}$ local diffeomorphism): for $C^{3}$ first-kind data with $\det S_{q Q} \neq 0$ the map defined by $p = S_{q} (q, Q)$ , $P = - S_{Q}$ is a canonical $C^{2}$ local diffeomorphism with unit determinant, and $p \cdot d q - P \cdot d Q = d S$ .

[F3]

Forms and calculus (The canonical one-form and the two-form of a potential presentation: $λ = p \cdot d x$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: $Φ_{X}^{*} λ = \sum P_{a} d q^{a}$ and $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ , Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests, A $C^{1}$ map between phase domains preserves the standard two-form iff $D C^{T} J D C = J$ , and a canonical diffeomorphism carries $X_{H \circ C}$ to $X_{H}$ , The pullback of a differential form, A smooth differential $k$ -form, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, Linear map between vector spaces over the same field, Invertible matrices and the general linear group ${GL}_{n} (F)$ , A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): the canonical one-form of the presentation is $λ = \sum_{a} p_{a} d q^{a}$ and pulls back as displayed, the canonical two-form is in canonical coordinates, and the algebraic and differential computations of the example use the ordinary product, chain and mixed-partial rules; a block matrix with invertible diagonal blocks is invertible.

Verification

technique · direct. Verify the explicit solution, compute the mixed Hessian and the generated map of $S_{t}$, check the projection matrix, and pull back the canonical one-form

1.1F1algebra

The displayed curve has $\dot{x} = m^{- 1} p$ and $\dot{p} = 0$ , hence solves the Hamiltonian system for all $t$ , so by the solutions clause of [F1] it is the free-particle flow; this is claim 1.

1.2F2F3algebra

Mixed Hessian and generated map: $S_{t, q} (q, Q) = - \frac{m}{2 t} \cdot 2 (Q - q) \cdot (- 1) = \frac{m}{t} (Q - q)$ , so $S_{t, q Q} = \frac{m}{t} I_{3}$ is invertible for $t \neq 0$ ; the first-kind relations of [F2] read $p = S_{t, q} = \frac{m}{t} (Q - q)$ and $P = - S_{t, Q} = \frac{m}{t} (Q - q) = p$ , hence $Q = q + \frac{t}{m} p$ and $P = p$ : the generated map is $Φ_{t}$ ; by [F2] it is canonical with unit determinant, which is claim 2.

2.1step 1.2F3algebra

Projection and degeneration: the derivative of $(q, p) \mapsto (q, Q)$ is the displayed block matrix with invertible diagonal blocks $I_{3}$ and $(t / m) I_{3}$ for $t \neq 0$ , so the projection is invertible there; at $t = 0$ the function $S_{t}$ is undefined and the lower-right block vanishes, so the projection is singular, and no member of the family represents the identity map; this is claim 3.

2.2step 1.2F2F3algebra

Pullback of the one-form: $Φ_{t}^{*} λ = P \cdot d Q$ with $Q = q + (t / m) p$ and $P = p$ , so $Φ_{t}^{*} λ = p \cdot (d q + (t / m) d p) = λ + (t / m) d (∣ p ∣^{2} / 2) = λ + t d K$ by the chain rule and the symmetry of the Euclidean product [F3]; differentiating gives $Φ_{t}^{*} d λ = d λ$ , and the matrix identity of claim 2 was already computed; this is claim 4.

3.1step 1.1step 1.2step 2.1step 2.2∎

Collecting steps 1.1, 1.2, 2.1 and 2.2 proves the four claims on the declared data.

The free particle of the model: the flow $Q = q + (t / m) p$ generated by the first-kind function $S_{t} (q, Q) = - m ∣ Q - q ∣^{2} / (2 t)$ , the projection check and the exact change of $λ$

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

The free particle of the model: the flow Q=q+(t/m)pQ=q+(t/m)p generated by the first-kind function St(q,Q)=−m∣Q−q∣2/(2t)St​(q,Q)=−m∣Q−q∣2/(2t), the projection check and the exact change of λλ

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

The free particle of the model: the flow $Q = q + (t / m) p$ generated by the first-kind function $S_{t} (q, Q) = - m ∣ Q - q ∣^{2} / (2 t)$ , the projection check and the exact change of $λ$