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Equal-dimensional canonical transformations of a potential presentation conjugate Hamiltonian fields and transport systems and brackets

Physical scope

Transformation theory for equal-dimensional local canonical transformations of the declared potential presentation: a $C^{2}$ local diffeomorphism preserving the two-form conjugates the Hamiltonian field of a phase function to the field of the composed function, preserves the model's Poisson bracket under composition, and carries solutions of one Hamiltonian system to solutions of the other, also for time-dependent Hamiltonians. No such ambient field-transport result is asserted for a lower-dimensional symplectic immersion. The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics; it asserts no empirical content or global existence.

Statement

(field transport) For every $C^{2}$ function $f$ on $W_{2}$ and every $z \in W_{1}$ ,

$D C (z) X_{f \circ C} (z) = X_{f} (C (z)),$

i.e. $C$ conjugates the Hamiltonian field of $f \circ C$ on $W_{1}$ to the field of $f$ on $W_{2}$ . 2. (bracket invariance) For all $C^{2}$ functions $f, g$ on $W_{2}$ ,

${f \circ C, g \circ C} = {f, g} \circ C on W_{1},$

so the bracket of the presentation is invariant under canonical maps. 3. (transport of solutions) Let $J \subseteq I_{0}$ be a nonempty open interval and let $z : J \to W_{1}$ be a $C^{1}$ curve. Then $z$ solves $\dot{z} (t) = X_{(H \circ C)_{t}} (z (t))$ if and only if $t \mapsto C (z (t))$ solves $\dot{w} (t) = X_{H_{t}} (w (t))$ on $J$ : the canonical map carries the Hamiltonian system of $H \circ C$ to that of $H$ , with time dependence allowed. 4. (instances) Genuine transitions between full-dimensional ( $r = 3 N$ ) charts of the presentation and the maps generated by first- and second-kind generating functions satisfy these transport conclusions. A fixed time-slice map $Φ_{t_{*}}$ of The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time is an additional instance for clauses 1-2 when it has $C^{2}$ regularity, with $Φ_{t_{*}}$ held fixed as the coordinate map. The flow-preservation theorem under its stated $C^{2}$ Hamiltonian hypothesis guarantees only $C^{1}$ regularity; this item makes no field-transport claim for a slice known only to be $C^{1}$ , and does not treat the moving family $t \mapsto Φ_{t}$ as a single time-dependent coordinate transformation.

The statement is conditional on the adopted force branch, the declared presentation and the cited mathematics; it asserts no empirical content and no global existence of canonical maps.

Facts & Assumptions

Given: The presentation with its two-form, canonical charts, fields and bracket, a $C^{2}$ canonical map $C : W_{1} \to W_{2}$ , a $C^{2}$ function $H$ on $I_{0} \times W_{2}$ , and a $C^{1}$ curve $z$ in $W_{1}$ .

[F1]

Canonical maps, fields and brackets (Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests, 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 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, 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 Poisson bracket of phase functions of a potential presentation: ${f, g} = ω (X_{f}, X_{g})$ and the coordinate formula in each canonical 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 $X_{H}$ , The Hamiltonian field of a $C^{2}$ function on a phase domain is $J \nabla H$ , and the Poisson bracket is $\sum_{i} (\partial_{q^{i}} f \partial_{p_{i}} g - \partial_{p_{i}} f \partial_{q^{i}} g)$ , A phase domain, its canonical one-form and its standard symplectic two-form, The bracket of a potential presentation is bilinear, skew, a derivation in each slot, satisfies the Jacobi identity, and reproduces the fundamental brackets and the Hamiltonian field): $C$ is $C^{2}$ between phase domains of the same dimension, is a local diffeomorphism and satisfies $C^{*} ω_{2} = ω_{1}$ , equivalently $D C (z)^{T} J_{n} D C (z) = J_{n}$ for every $z$ , so $D C (z)$ is invertible and $D C (z) J_{n} D C (z)^{T} = J_{n}$ ; the field of a $C^{2}$ function $f$ is $X_{f} = J_{n} \nabla f$ in a canonical chart and satisfies $D C X_{f \circ C} = X_{f} \circ C$ ; the bracket is ${f, g} = \nabla f^{T} J_{n} \nabla g$ .

[F2]

Chain rule and matrix algebra (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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, 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, Linear map between vector spaces over the same field, $C^{k}$ Euclidean maps and diffeomorphisms): $\nabla (f \circ C) = D C^{T} (\nabla f) \circ C$ and $(C \circ z)^{'} (t) = D C (z (t)) \dot{z} (t)$ on curves, by the chain rule; products, transposes and inverses of the displayed matrices obey the ordinary identities, and $J^{- 1} = - J$ .

[F3]

Curves and equations (First-order systems, initial value problems, and solutions on intervals, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, 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): a $C^{1}$ curve solves the Hamiltonian system of the given $C^{2}$ function $H$ exactly when its velocity equals $X_{H_{t}}$ at its points. Identification with Cartesian model motions on $d \equiv 0$ requires the mechanical branch Hamiltonian $H (t, x, p) = \frac{1}{2} p \cdot M^{- 1} p + U (t, x)$ ; its first equation gives $p = M x^{'}$ and hence $x \in C^{2}$ . For the generalized-coordinate mechanical Hamiltonian, clause 2 of the cited theorem retains the residual generalized force $Q^{r e s}$ in the momentum equation. Thus its integral curves coincide with the declared Lagrange equations only when $Q^{r e s}$ vanishes along the curve, in particular on $d \equiv 0$ ; otherwise those equations are the forced Hamilton system. No identification with model motions is asserted for an arbitrary $H$ .

[F4]

Generated and chart 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, The Euclidean inverse function theorem, 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 $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ ): first- and second-kind data are $C^{3}$ functions $S$ on nonempty open product domains with invertible mixed Hessian $S_{q Q}$ or $S_{q P}$ . They define $p = S_{q}$ , $P = - S_{Q}$ or $p = S_{q}$ , $Q = S_{P}$ by local inversion of the relation map $Ψ = (q, S_{q})$ . The mathematical proposition proves their standard-form pullback identity. Its $C^{2}$ inverse regularity follows directly: the $C^{1}$ local inverse $g$ satisfies $D g = (D Ψ \circ g)^{- 1}$ , which is $C^{1}$ by the cofactor-over-determinant formula, since $D Ψ$ and $g$ are $C^{1}$ . Hence the generated map is $C^{2}$ . These maps are instances here on branches with source and target in the declared presentation domains, as required by the given $C : W_{1} \to W_{2}$ ; the mathematical proposition does not assert physical-domain containment. Genuine full-dimensional momentum-chart transitions are $C^{2}$ and preserve the source and target forms by clauses 2-3 of the two-form theorem.

Proof

technique · direct. Transport the matrix identity to $DC\,J\,DC^{\mathsf T}=J$, compute the bracket of composed functions, and carry the field equation along the map by the chain rule

1.1F1F2algebra

By [F1] the map $C$ satisfies $D C^{T} J_{n} D C = J_{n}$ everywhere and $D C$ is invertible; taking inverses of the identity gives $D C^{- 1} J_{n}^{- 1} D C^{- T} = J_{n}^{- 1}$ , that is $D C^{- 1} J_{n} D C^{- T} = J_{n}$ because $J_{n}^{- 1} = - J_{n}$ , and multiplying on the left by $D C$ and on the right by $D C^{T}$ gives $D C J_{n} D C^{T} = J_{n}$ .

1.2F1

By [F1] the fields satisfy $D C X_{f \circ C} = X_{f} \circ C$ for every $C^{2}$ function $f$ on $W_{2}$ ; this is clause 1.

2.1step 1.1F1F2algebra

Bracket invariance: for $C^{2}$ functions $f, g$ on $W_{2}$ , the chain rule of [F2] gives $\nabla (f \circ C) (z) = D C (z)^{T} \nabla f (C (z))$ and $\nabla (g \circ C) (z) = D C (z)^{T} \nabla g (C (z))$ , so ${f \circ C, g \circ C} (z) = \nabla f (C (z))^{T} D C (z) J D C (z)^{T} \nabla g (C (z)) = \nabla f (C (z))^{T} J \nabla g (C (z)) = {f, g} (C (z))$ by step 1.1 and the bracket formula of [F1]; this is clause 2.

2.2step 1.2F2F3algebra

Transport of solutions, forward direction: if $z$ solves $\dot{z} = X_{H_{t} \circ C} (z)$ , then $w : = C \circ z$ has $\dot{w} = D C (z) \dot{z} = D C (z) X_{H_{t} \circ C} (z) = X_{H_{t}} (C (z))$ by the chain rule of [F2] and the field transport of step 1.2, so $w$ solves the $H$ -system.

3.1step 1.2step 2.2F1F2F4

Transport of solutions, converse: if $w = C \circ z$ solves the $H$ -system, then $\dot{w} = X_{H_{t}} (w)$ and, $D C (z)$ being invertible by [F1], $\dot{z} = D C (z)^{- 1} \dot{w} = D C (z)^{- 1} X_{H_{t}} (C (z)) = X_{H_{t} \circ C} (z)$ by step 1.2 applied to $D C X_{H_{t} \circ C} = X_{H_{t}} \circ C$ ; hence $z$ solves the system of $H \circ C$ , proving clause 3 in both directions. For clause 4, [F4] supplies the $C^{2}$ regularity and canonicity of genuine full-dimensional chart transitions and first- and second-kind generated maps on their admissible presentation domains. Applying clauses 1-3 to each such fixed map gives the stated instances. A flow slice is included only as a fixed map when it has that same $C^{2}$ regularity; the moving time-dependent family is not covered by this fixed-map theorem.

4.1step 1.1step 1.2step 2.1step 2.2step 3.1∎

Collection: clause 1 is steps 1.1-1.2, clause 2 is step 2.1, clause 3 is steps 2.2 and 3.1, and clause 4 applies to the stated full-dimensional transitions and generating-function maps, plus any flow slice that independently has the required $C^{2}$ regularity; this proves the theorem.

Remarks

Why the bracket is invariant. The proof uses the equivalent form $D C J D C^{T} = J$ of the canonical matrix identity, which is not automatic from $\det D C = 1$ in dimension $2 m \geq 4$ ; the companion page's volume-preserving shear is a counterexample to bracket invariance for a map satisfying only $∣ \det D C ∣ = 1$ .
No global claims. The transport is asserted on the domain of the canonical map; the existence of a canonical map with specified data is the subject of 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 $K = H + \partial_{t} S$ and is local in general.

Equal-dimensional canonical transformations of a potential presentation conjugate Hamiltonian fields and transport systems and brackets

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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