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The point transformation of the model: the chart transition $(q, π) \mapsto (X (q), D X (q)^{- T} π)$ is generated in the second kind, has a nowhere invertible $(q, Q)$ -projection, and is contrasted with the momentum shift

Physical scope

A worked projection and generating-function computation for the declared potential presentation: a genuine chart $X$ of the configuration domain induces the momentum chart transition from generalized momentum $π$ to Cartesian momentum $P$ , $(q, π) \mapsto (X (q), D X (q)^{- T} π)$ , generated in the second kind by the action-dimensioned function $S (q, P) = P \cdot X (q)$ ; the transition has unit determinant and fails the first-kind projection hypothesis identically. The degenerate case is compared with the momentum shift, whose second-kind generating function is global. 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 $m > 0$ , configuration domain $Q = R^{3}$ , a $C^{2}$ potential and no residual forces, and let $X : Ω \to Q$ be a generalized-coordinate chart of the configuration domain: $r = 3$ , $X$ of class $C^{3}$ , $D X (q)$ invertible for every $q \in Ω$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, 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). Let the generalized coordinates have declared dimensions $[q^{a}] = γ_{a}$ ; their conjugate generalized momenta $π_{a} = \partial \tilde{L} / \partial {\dot{q}}^{a}$ have dimensions $[π_{a}] = μ ℓ^{2} τ^{- 1} γ_{a}^{- 1}$ . The Cartesian positions $X^{i} (q)$ have dimension $ℓ$ and their conjugate Cartesian momenta $P_{i}$ have dimension $μ ℓ τ^{- 1}$ , by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework and Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces. The induced momentum chart transition of the presentation is

$Φ_{X} (q, π) = (X (q), M D X (q) G (q)^{- 1} π) = (X (q), D X (q)^{- T} π),$

the second equality because $M D X G^{- 1} = D X^{- T}$ when $D X$ is invertible and $G = D X^{T} M D X$ (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 momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Hyperregular potential presentations and their inverse Legendre branches in the model's charts). Then:

the function $S (q, P) : = P \cdot X (q) = \sum_{i = 1}^{3} P_{i} X^{i} (q)$ on $Ω \times R^{3}$ is of class $C^{3}$ and has action dimension $[S] = [P_{i}] [X^{i}] = μ ℓ^{2} τ^{- 1}$ ; its mixed derivative matrix $S_{q P}$ is $D X (q)^{T}$ , whose determinant is nonzero. Here $P$ is the new Cartesian momentum, while $π$ is the old generalized momentum. The second-kind relations are $π = S_{q} = D X (q)^{T} P$ and $Q = S_{P} = X (q)$ , so solving for $P$ gives $(q, π) \mapsto (X (q), D X (q)^{- T} π)$ , exactly $Φ_{X}$ ; consequently $Φ_{X}$ is a canonical map of the presentation with $∣ \det D Φ_{X} ∣ = 1$ (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);
determinant check: with the block decomposition in source coordinates $(q, π)$ and target coordinates $(Q, P)$ the derivative is block triangular,

$D Φ_{X} = (\begin{array}{cc} D X (q) & 0 \\ \partial_{q} (D X^{- T} π) & D X (q)^{- T} \end{array}), \det D Φ_{X} = \det D X (q) \cdot \det D X (q)^{- T} = 1;$

the $(q, Q)$ -projection $(q, π) \mapsto (q, X (q))$ is independent of the momentum and has derivative $(\begin{array}{cc} I_{3} & 0 \\ D X (q) & 0 \end{array})$ of rank $3 < 6$ at every point: it is nowhere invertible. Since a first-kind generating function requires the relation $p = S_{q} (q, Q)$ to be solvable for $Q$ , i.e. the map $(q, Q) \mapsto (q, S_{q} (q, Q))$ to be a local diffeomorphism, the transition $Φ_{X}$ admits no first-kind generating function near any point; the second-kind representation of claim 1 is the available local description (Generating functions of the first and second kind for a local canonical map, A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function);
contrast — the Cartesian momentum shift: for a constant $c \in R^{3}$ of momentum dimension the map $C (q, p) = (q, p + c)$ of the Cartesian chart is canonical trivially ( $D C = I_{6}$ ), its first-kind projection is degenerate identically, and it has the global second-kind generating function $S (q, P) = q \cdot (P - c)$ , also of action dimension, since $S_{q} = P - c$ and $S_{P} = q$ (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 $K = H + \partial_{t} S$ clause 2).

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

Facts & Assumptions

Given: The one-label model with a genuine chart $X$ of the configuration domain, the induced momentum chart transition $Φ_{X}$ , and the generating-function data of the two examples.

[F1]

Chart data and units (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Hyperregular potential presentations and their inverse Legendre branches in the model's charts, 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 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 $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ , 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): $X$ is a $C^{3}$ immersion with $D X (q)$ invertible, $G = D X^{T} M D X$ , the generalized momentum is $π = G (q) \dot{q}$ with $[π_{a}] = μ ℓ^{2} τ^{- 1} γ_{a}^{- 1}$ , and the transition is $Φ_{X} (q, π) = (X (q), M D X (q) G (q)^{- 1} π) = (X (q), D X (q)^{- T} π)$ , whose output is Cartesian momentum of dimension $μ ℓ τ^{- 1}$ .

[F2]

Generating functions and their canonical maps (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, A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function, The Euclidean inverse function theorem): a second-kind datum with $\det S_{q P} \neq 0$ generates a canonical $C^{2}$ local diffeomorphism via old generalized momentum $π = S_{q}$ and new Cartesian position $Q = S_{P}$ , with unit determinant; a first-kind datum requires $\det S_{q Q} \neq 0$ , so $(q, Q) \mapsto (q, S_{q} (q, Q))$ is a local diffeomorphism. The generating functions have action dimension. For the point transformation the mathematical proposition applies with $O_{q} = Ω$ and $O_{P} = R^{3}$ , using the standard source and target forms supplied by [F1]; the Cartesian-domain theorem 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 $K = H + \partial_{t} S$ clause 2 applies to the momentum shift with $O_{q} = O_{Q} = O_{P} = R^{3}$ .

[F3]

Calculus and matrix algebra ( $C^{k}$ 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 $U \subseteq R^{m} \to R^{n}$ , 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, Submersions and immersions between Euclidean open sets): $S_{q} = D X^{T} P$ has generalized-momentum components $[π_{a}] = [P_{i}] [X^{i}] [q^{a}]^{- 1}$ , so $S = P \cdot X (q)$ has action dimension; the displayed partial derivatives are computed by the ordinary rules, transposes and inverses obey $(A^{T})^{- 1} = (A^{- 1})^{T}$ , and a matrix with a vanishing block column has rank at most the number of its nonzero columns.

Verification

technique · direct. Compute the mixed Hessian of the second-kind datum, generate the map, check the block-triangular determinant, examine the rank of the projection, and generate the momentum shift

1.1F1F2F3algebra

Second-kind generation: by [F3], $S_{P_{i}} = X^{i} (q)$ and $S_{q^{a}} = \sum_{i} P_{i} \partial X^{i} / \partial q^{a}$ , so $S_{q P} = D X (q)^{T}$ has nonzero determinant by [F1]. The generated relations of [F2] are $π = D X (q)^{T} P$ and $Q = X (q)$ , i.e. $P = D X (q)^{- T} π$ and $Q = X (q)$ : this is $Φ_{X}$ , canonical with unit determinant and with $[S] = [P_{i} X^{i}] = μ ℓ^{2} τ^{- 1}$ ; claim 1.

1.2F2F3algebra

Momentum shift: for $C (q, p) = (q, p + c)$ the derivative is the identity, so $D C^{T} J D C = J$ and the map is canonical; with $S (q, P) = q \cdot (P - c)$ one has $S_{q} = P - c$ and $S_{P} = q$ , so [F2] gives $p = P - c$ , $Q = q$ , i.e. $C$ , with $\det S_{q P} = \det I_{3} = 1$ ; the first-kind projection $(q, p) \mapsto (q, q)$ is constant in $p$ and hence degenerate; this is claim 4.

2.1step 1.1F3algebra

Determinant check: in the coordinates $(q, π)$ on the source and $(Q, P)$ on the target, the derivative of $Φ_{X}$ is the displayed block-triangular matrix with diagonal blocks $D X (q)$ and $D X (q)^{- T}$ ; because $\det A^{T} = \det A$ and $\det (A^{- 1}) = (\det A)^{- 1}$ , its determinant is $\det D X \cdot (\det D X)^{- 1} = 1$ , which is claim 2.

2.2step 1.1F2F3algebra

Projection obstruction: the $(q, Q)$ -projection of $Φ_{X}$ sends $(q, π)$ to $(q, X (q))$ , with derivative the displayed block matrix whose second block column vanishes; its rank is at most $3 < 6$ , so it is nowhere invertible. By [F2], a first-kind generating function would make $(q, Q) \mapsto (q, S_{q} (q, Q))$ a local diffeomorphism whose inverse is exactly this projection, which is impossible; hence no first-kind generating function exists near any point, proving claim 3.

3.1step 1.1step 2.1step 2.2step 1.2∎

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

The point transformation of the model: the chart transition $(q, π) \mapsto (X (q), D X (q)^{- T} π)$ is generated in the second kind, has a nowhere invertible $(q, Q)$ -projection, and is contrasted with the momentum shift

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

The point transformation of the model: the chart transition (q,π)↦(X(q),DX(q)−Tπ)(q,π)↦(X(q),DX(q)−Tπ) is generated in the second kind, has a nowhere invertible (q,Q)(q,Q)-projection, and is contrasted with the momentum shift

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources

The point transformation of the model: the chart transition $(q, π) \mapsto (X (q), D X (q)^{- T} π)$ is generated in the second kind, has a nowhere invertible $(q, Q)$ -projection, and is contrasted with the momentum shift