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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 $K = H + \partial_{t} S$

Physical scope

Generating-function theory for the declared potential presentation: first- and second-kind generating functions of action dimension generate equal-dimensional canonical local diffeomorphisms with the recorded one-form identities and unit determinant; a canonical diffeomorphism with invertible $(q, Q)$ -projection is locally generated in the first kind; and for a time-dependent generating function the time derivative has energy dimension and the transformed Hamiltonian is $K = H + \partial_{t} S$ , with the Hamilton systems of $H$ and $K$ carried into one another by the generated time-dependent map. The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics; it asserts no global generating function and no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions with the momentum charts and two-form datum of 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 canonical coordinates of Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts, the Hamiltonian fields of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian, the canonical maps of Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests and the branch Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions and 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 (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). Write $n^{'} : = 3 N$ . Let $O_{q}, O_{Q} \subseteq Q$ be nonempty open Cartesian position-coordinate sets, and let $O_{P} \subseteq R^{n^{'}}$ be a nonempty open momentum-coordinate set. In clause 2 require $S_{P} (q, P) \in O_{Q}$ for every $(q, P) \in O_{q} \times O_{P}$ ; only local branches with source in $O_{q} \times R^{n^{'}}$ and target in $O_{Q} \times R^{n^{'}}$ are called maps of the presentation. Then:

(first kind) Let $S : O_{q} \times O_{Q} \to R$ be of class $C^{3}$ with $\det S_{q Q} (q, Q) \neq 0$ everywhere (Generating functions of the first and second kind for a local canonical map), and define the local map generated by $S$ by solving $p = S_{q} (q, Q)$ for $Q = Q (q, p)$ and setting $P : = - S_{Q} (q, Q)$ : a $C^{2}$ map $C : (q, p) \mapsto (Q, P)$ defined on a neighbourhood of each point of the solution set. Then $C$ is a canonical $C^{2}$ local diffeomorphism of the presentation: $D C^{T} J D C = J$ , $C^{*} ω = ω$ and $∣ \det D C ∣ = 1$ ; and the one-form identity $p \cdot d q - P \cdot d Q = d S$ holds on the $(q, Q)$ -domain (The local map generated by a first- or second-kind generating function is a canonical $C^{2}$ local diffeomorphism).
(second kind) Let $S : O_{q} \times O_{P} \to R$ be $C^{3}$ with $\det S_{q P} \neq 0$ everywhere, and define the local map by $p = S_{q} (q, P)$ , $Q = S_{P} (q, P)$ . Then it is again a canonical $C^{2}$ local diffeomorphism of the presentation with the recorded identity $p \cdot d q + Q \cdot d P = d S$ .
(local converse) Let $C$ be a $C^{2}$ canonical diffeomorphism between nonempty open subsets of the presentation's Cartesian phase domains, and suppose that the projection $z \mapsto (q (z), Q (z))$ onto the old position and the new position has invertible derivative at a point $z_{0}$ . Then there are a neighbourhood of $z_{0}$ and a $C^{3}$ function $S : O_{q} \times O_{Q} \to R$ with $\det S_{q Q} \neq 0$ whose generated map agrees with $C$ on that neighbourhood (A canonical map with invertible $(q, Q)$ -projection is locally generated by a first-kind generating function).
(time-dependent generating function) Let $J \subseteq I_{0}$ be a nonempty open interval and let $S : J \times O_{q} \times O_{Q} \to R$ be of class $C^{3}$ with $\det S_{q Q} (t, q, Q) \neq 0$ everywhere; for each $t$ let $C_{t}$ be the first-kind map generated by $S_{t}$ . Assume the values $(q, p)$ used below lie in the domain of $H$ of The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian, and set $p : = S_{q} (t, q, Q)$ , $P : = - S_{Q} (t, q, Q)$ and $K (t, q, Q) : = H (t, q, S_{q} (t, q, Q)) + \partial_{t} S (t, q, Q)$ . The functions $S$ in clauses 1-4 have action dimension $μ ℓ^{2} τ^{- 1}$ ; in clause 4, $\partial_{t} S$ has energy dimension $μ ℓ^{2} τ^{- 2}$ , so the sum defining $K$ is dimensionally consistent. Then: (a) each $C_{t}$ is a canonical $C^{2}$ local diffeomorphism of the presentation; (b) the Poincare-Cartan identity

$p \cdot d q - H d t = P \cdot d Q - K d t + d S$

holds on $J \times O_{q} \times O_{Q}$ (A time-dependent generating function produces canonical time slices and shifts the Hamiltonian by $S_{t}$ ); and (c) writing $K$ as a function of $(t, Q, P)$ through the correspondence $(t, q, Q) \leftrightarrow (t, Q, P)$ , a $C^{1}$ curve $t \mapsto (q (t), p (t))$ satisfies $q^{'} = H_{p}$ , $p^{'} = - H_{q}$ if and only if $t \mapsto (Q (t), P (t)) = C_{t} (q (t), p (t))$ satisfies $Q^{'} = K_{P}$ , $P^{'} = - K_{Q}$ on the interval where both descriptions are defined.

The statement is conditional on the adopted force branch, the declared presentation and the cited mathematics; it asserts no global generating function, no existence of a canonical map with prescribed data beyond the local statements, and no empirical content.

Facts & Assumptions

Given: The presentation with its Cartesian two-form and canonical charts, the generating-function data $S$ , the generated maps $C$ resp. $C_{t}$ , and the functions $p, P, K$ of the item.

[F1]

The presentation's two-form (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 coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts, Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests): on the Cartesian chart the two-form is $ω = \sum_{a} d q^{a} \land d p_{a}$ with standard matrix $J$ ; a $C^{1}$ map between phase domains is canonical exactly when it satisfies the matrix identity, and then it preserves $ω$ .

[F2]

Generating functions and their 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): when $S$ is $C^{3}$ , each defining relation map, such as $Ψ (q, Q) = (q, S_{q} (q, Q))$ , is $C^{2}$ and has invertible derivative under the mixed-Hessian hypothesis. The cited inverse-function theorem gives a $C^{1}$ local inverse $g$ and the derivative formula $D g (y) = [D Ψ (g (y))]^{- 1}$ . Since $D Ψ$ and $g$ are $C^{1}$ , this right side is $C^{1}$ : matrix inversion is $C^{1}$ on invertible matrices by the cofactor-over-determinant formula. Hence $g$ is $C^{2}$ , and composition in the generating-function definitions gives $C^{2}$ generated local maps; the same reasoning applies to second-kind and parameter-dependent relation maps. The generating-function propositions supply their canonical identities, determinants and one-form identities. A canonical $C^{2}$ diffeomorphism with invertible $(q, Q)$ -projection is locally generated in the first kind.

[F3]

Time-dependent generating functions (A time-dependent generating function produces canonical time slices and shifts the Hamiltonian by $S_{t}$ ): for a $C^{3}$ family $S (t, q, Q)$ with $\det S_{q Q} \neq 0$ the maps $C_{t}$ are canonical local diffeomorphisms, and on $J \times O_{q} \times O_{Q}$ the identity $p d q - H d t = P d Q - K d t + d S$ holds with $K = H + \partial_{t} S$ .

[F4]

Coordinate Poincare-Cartan forms and transport (A curve solves Hamilton's equations if and only if its velocity annihilates $d (p \cdot d q - H d t)$ , with the three coordinate components computed, The pullback of a differential form, A smooth differential $k$ -form, The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian, Equal-dimensional canonical transformations of a potential presentation conjugate Hamiltonian fields and transport systems and brackets): for a $C^{1}$ Hamiltonian the lemma defines the continuous coordinate two-form from the first derivatives of $p d q - F d t$ and characterizes Hamilton solutions by annihilation. For a $C^{2}$ coordinate map $Γ$ and a $C^{1}$ one-form $α = \sum_{j} a_{j} d u^{j}$ , the coefficient formula for $Γ^{*} α$ and equality of mixed second derivatives give $d (Γ^{*} α) = Γ^{*} (d α)$ : the second-derivative terms in $\partial_{i} [(a_{j} \circ Γ) \partial_{k} Γ^{j}] - \partial_{k} [(a_{j} \circ Γ) \partial_{i} Γ^{j}]$ cancel, leaving the pointwise pullback of $d α$ . Equal-dimensional canonical local maps conjugate the Hamiltonian systems by the transport theorem.

[F5]

Chart calculus ( $C^{k}$ 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, 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, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): the maps displayed above are $C^{2}$ or $C^{1}$ according to their displayed formulae, the inverse-function theorem applies wherever the mixed Hessian determinant is nonzero, and matrix products, transposes and inverses obey the ordinary rules.

Proof

technique · direct. Instantiate the generating-function theorems on the presentation's Cartesian chart for clauses 1-3, then carry the time-dependent Poincare-Cartan identity to the two Hamilton systems by the annihilation characterization

1.1F1F2algebra

Clauses 1 and 2: in the first-kind case the relation map $Ψ (q, Q) = (q, S_{q} (q, Q))$ is $C^{2}$ and has invertible derivative by $\det S_{q Q} \neq 0$ ; in the second-kind case use $Ψ (q, P) = (q, S_{q} (q, P))$ and $\det S_{q P} \neq 0$ . For either map, the inverse theorem gives a local $C^{1}$ inverse $g$ and $D g (y) = [D Ψ (g (y))]^{- 1}$ . Since $D Ψ$ and $g$ are $C^{1}$ , this derivative is $C^{1}$ , because the cofactor-over-determinant formula makes matrix inversion $C^{1}$ on the invertible matrices; thus $g$ is $C^{2}$ . The maps in the generating-function definitions are then compositions of $C^{2}$ maps, so the generated maps are $C^{2}$ local diffeomorphisms. The stated position containment puts the source and target of each branch inside the presentation's open phase domain. With [F1] identifying the presentation's Cartesian chart with the standard phase domain and $n^{'} = 3 N$ , the canonical identity, matrix test, unit determinant and one-form identities follow from [F2].

1.2F2algebra

Clause 3: let $F (z) = (q (z), Q (z))$ be the stated $(q, Q)$ projection. Since $C$ is $C^{2}$ , $F$ is $C^{2}$ , and its derivative is invertible at the base point by hypothesis. The inverse theorem gives a $C^{1}$ local inverse $G$ ; differentiating $F \circ G = i d$ gives $D G = (D F \circ G)^{- 1}$ . As $D F$ and $G$ are $C^{1}$ , the right side is $C^{1}$ by the cofactor-over-determinant formula for matrix inversion, so $G$ is $C^{2}$ . This supplies the $C^{2}$ inverse regularity needed in the proof of the local-converse proposition [F2]. Its remaining argument then yields a neighborhood and a $C^{3}$ first-kind generating function with $\det S_{q Q} \neq 0$ whose generated map agrees with $C$ , proving clause 3.

1.3F1F3

Clause 4(a) and (b): by [F3] each $C_{t}$ is a canonical $C^{2}$ local diffeomorphism and the identity $p d q - H d t = P d Q - K d t + d S$ holds with $K = H + \partial_{t} S$ ; this is clause 4(a)-(b).

2.1step 1.3F2F4F5

Clause 4(c), setup: the relation maps $Φ : (t, q, Q) \mapsto (t, q, p)$ and $Λ : (t, q, Q) \mapsto (t, Q, P)$ with $p = S_{q}$ , $P = - S_{Q}$ are $C^{2}$ local diffeomorphisms: their derivatives are invertible by $\det S_{q Q} \neq 0$ , the inverse theorem gives $C^{1}$ inverses, and the derivative formula in [F2] upgrades those inverses to $C^{2}$ . Pull the two Poincare-Cartan one-forms back by the pointwise coordinate formula; their resulting $C^{2}$ one-forms on the parameter domain are the two sides $α_{H}$ and $α_{K}$ of the identity in clause 4(b), so $α_{H} = α_{K} + d S$ . Applying the coordinate derivative to this identity gives $d α_{H} = d α_{K}$ , since $d (d S) = 0$ by equality of mixed partials of $S$ . By the direct $C^{2}$ coordinate calculation in [F4], these derivatives equal $Φ^{*} σ_{H}$ and $Λ^{*} σ_{K}$ , respectively, where $σ_{H} = d p \land d q - d H \land d t$ and $σ_{K} = d P \land d Q - d K \land d t$ are the continuous coordinate two-forms of the lemma. Thus $Φ^{*} σ_{H} = Λ^{*} σ_{K}$ .

3.1step 2.1F4algebra

Clause 4(c), forward direction: let $t \mapsto (q (t), p (t))$ solve the $H$ -system on an interval and define $Q (t)$ by $p (t) = S_{q} (t, q (t), Q (t))$ , $P (t) : = - S_{Q} (t, q (t), Q (t))$ ; then $w : = (1, q^{'}, Q^{'})$ is the velocity of the graph curve and $D Φ w = (1, q^{'}, p^{'})$ , $D Λ w = (1, Q^{'}, P^{'})$ by the chain rule, while $ι_{(1, q^{'}, p^{'})} σ_{H} = 0$ by [F4]; for every tangent vector $u$ at the corresponding point, $σ_{H} (D Φ w, D Φ u) = (Φ^{*} σ_{H}) (w, u) = (Λ^{*} σ_{K}) (w, u) = σ_{K} (D Λ w, D Λ u)$ , and since $D Λ$ is invertible and $D Λ u$ ranges over all tangent vectors, $ι_{(1, Q^{'}, P^{'})} σ_{K} = 0$ ; by [F4] the curve $(Q, P)$ solves the $K$ -system.

3.2step 2.1F2F4F5algebra

Clause 4(c), converse: if $(Q (t), P (t))$ solves the $K$ -system, solve $- S_{Q} (t, q, Q (t)) = P (t)$ locally for $q (t)$ by [F2, F5], since its derivative in $q$ is $- S_{Q q} = - S_{q Q}^{T}$ and is invertible; then set $p (t) : = S_{q} (t, q (t), Q (t))$ , so that the correspondence is inverted; the same computation with the roles of $Φ$ and $Λ$ interchanged and the identity $Λ^{*} σ_{K} = Φ^{*} σ_{H}$ gives $ι_{(1, q^{'}, p^{'})} σ_{H} = 0$ , hence by [F4] the curve $(q, p)$ solves the $H$ -system.

4.1step 1.1step 1.2step 1.3step 3.1step 3.2∎

Collection: clauses 1-2 are step 1.1, clause 3 is step 1.2, clause 4(a)-(b) is step 1.3 and clause 4(c) is steps 3.1-3.2; this proves the theorem.

Remarks

The projection hypothesis is essential. The first-kind representation requires the invertibility of $Q \mapsto S_{q} (q, Q)$ , i.e. of the $(q, Q)$ projection of the canonical map; without it only other kinds are available locally. The companion page carries a point transformation whose $(q, Q)$ -projection is degenerate identically and which is nevertheless generated in the second kind.
No global generating function. The theorem is local. A canonical map of the presentation need not admit a globally defined generating function even when it is symplectic; the obstruction is a cohomological one and is described in Scope of the symplectic and canonical methods for a potential presentation: locality of generating functions, projections and determinant tests, absent global claims, and empirical status.

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 $K = H + \partial_{t} S$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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+∂tSK=H+∂t​S

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

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 $K = H + \partial_{t} S$