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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

Physical scope

The algebraic identities of the Poisson bracket of a declared potential presentation: bilinearity, skew-symmetry, the Leibniz rule in each slot, the Jacobi identity for $C^{2}$ phase functions through the pointwise C1-slot extension, the smooth Poisson algebra, the identity ${f, g} = X_{g} (f)$ relating the bracket to the presentation's Hamiltonian fields, and the fundamental brackets of the model's canonical coordinates. The statement is a conditional deduction from the adopted force branch, the declared presentation and the cited mathematics; it asserts no equation of motion, no conservation law and no measured value.

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 and the bracket of 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 on a nonempty open subset $W_{0}$ of a phase domain of the presentation. Then:

(bilinearity and skew-symmetry) The bracket is real-bilinear: ${α f + β g, h} = α {f, h} + β {g, h}$ and ${f, α g + β h} = α {f, g} + β {f, h}$ for real $α, β$ and $C^{2}$ phase functions; it is skew-symmetric, ${f, g} = - {g, f}$ , and in particular ${f, f} = 0$ .
(Leibniz rule) For $C^{2}$ functions $f, g, h$ ,

${f, g h} = {f, g} h + g {f, h}, {f g, h} = f {g, h} + g {f, h} .$

(Jacobi identity) For $C^{2}$ functions $f, g, h$ ,

${f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0.$

(relation to the fields) For $C^{2}$ functions $f, g$ one has ${f, g} = X_{g} (f) = - X_{f} (g) = d ⁣ f (X_{g}) = - d ⁣ g (X_{f})$ , and the bracket is $C^{1}$ ; in the Cartesian canonical coordinates ${f, g} = \sum_{a} (f_{x^{a}} g_{p_{a}} - f_{p_{a}} g_{x^{a}}) = \nabla f^{T} J \nabla g$ , and in any canonical chart $(θ^{1}, \dots, θ^{2 m})$ the analogous formula of 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 clause (c) holds.
(fundamental brackets) In the Cartesian canonical coordinates the coordinate functions satisfy ${x^{a}, x^{b}} = 0$ , ${p_{a}, p_{b}} = 0$ and ${x^{a}, p_{b}} = δ_{b}^{a}$ ; more generally, in any canonical chart $(θ^{1}, \dots, θ^{2 m})$ one has ${θ^{i}, θ^{j}} = J^{i j}$ , the entries of the standard symplectic matrix.

For nested brackets use the pointwise $C^{1}$ -slot coordinate extension of Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts, which agrees with the presentation bracket on $C^{2}$ inputs. Thus the Jacobi identity above is meaningful for $C^{2}$ functions despite their brackets being only $C^{1}$ . The $C^{2}$ class is not in general closed under the bracket and is not asserted to be a Poisson algebra. The smooth $C^{\infty}$ phase functions, equipped with this bracket and pointwise product, form a Poisson algebra. The statement is conditional on the adopted force branch, the declared presentation and the cited mathematics; it asserts no equation of motion, no conservation law and no measured value.

Facts & Assumptions

Given: The presentation with its two-form, canonical charts, Hamiltonian fields and bracket on the open phase domain $W_{0}$ .

[F1]

Bracket and fields (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, 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 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 $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ ): ${f, g} = ω (X_{f}, X_{g}) = d f (X_{g})$ for $C^{2}$ functions, the fields are the unique $C^{1}$ fields with $ι_{X_{f}} ω = d f$ , and the canonical charts carry the canonical coordinate expression of $ω$ .

[F2]

Coordinate calculus and the standard form (The Hamiltonian field of a $C^{2}$ function on a phase domain is $J \nabla H$ , and the Poisson bracket is $\sum_{i} (\partial_{q^{i}} f \partial_{p_{i}} g - \partial_{p_{i}} f \partial_{q^{i}} g)$ , Poisson bracket on a symplectic manifold, The Hessian of a $C^{2}$ scalar field is symmetric, The Hessian matrix and critical points of a scalar field, 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 chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ): in a phase chart the Hamiltonian field is $J \nabla f$ , the bracket is ${f, g} = \nabla f^{T} J \nabla g = \sum_{a} (f_{q^{a}} g_{p_{a}} - f_{p_{a}} g_{q^{a}})$ and is $C^{1}$ ; second partial derivatives of $C^{2}$ functions are symmetric at every point; scalar and matrix products and partial derivatives obey the ordinary product, sum and chain rules.

[F3]

Algebra of forms and matrices (The exterior derivative is a graded derivation, 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, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms, A smooth differential $k$ -form): the exterior derivative satisfies $d (g h) = g d h + h d g$ ; $J^{T} = - J$ , $J^{2} = - I$ ; transposes, products and Euclidean pairings obey the ordinary identities; $C^{2}$ and $C^{3}$ classes are closed under the displayed algebra.

Proof

technique · direct. Read the bracket from its coordinate formula, obtain the Leibniz rule from the derivation property of $d$, prove Jacobi by expanding into Hessian terms and cancelling cyclic groups by transposition, and evaluate the coordinate functions for the fundamental brackets

1.1F1F2algebra

By [F1] and [F2] the bracket satisfies ${f, g} = d f (X_{g}) = - d g (X_{f})$ , and $α X_{f} + β X_{g} = X_{α f + β g}$ because $ι_{X} ω = d (\cdot)$ is linear and $ω$ is bilinear; since $ω$ is antisymmetric, ${f, g} = ω (X_{f}, X_{g}) = - ω (X_{g}, X_{f}) = - {g, f}$ and ${f, f} = 0$ ; the coordinate formula of [F2] displays the same statements, proving clause 1 and the first identity of clause 4.

1.2F2F3algebra

Hessian lemma: for $C^{2}$ functions $u, v$ , the gradient of ${u, v}$ is $H e s s (u) J \nabla v - H e s s (v) J \nabla u$ ; indeed the $i$ -th partial derivative of $(\nabla u)^{T} J \nabla v$ is $\sum_{k, l} (u_{z^{k} z^{i}} J^{k l} v_{z^{l}} + u_{z^{k}} J^{k l} v_{z^{l} z^{i}})$ , and by symmetry of the Hessians [F2] and $J^{T} = - J$ [F3] this equals the $i$ -th component of $H e s s (u) J \nabla v - H e s s (v) J \nabla u$ .

1.3F1F2algebra

Fundamental brackets: in a canonical chart the two-form is $ω = \sum_{a} d θ^{a} \land d θ^{m + a}$ by [F1], and the coordinate formula of [F2] applied to the coordinate functions gives ${θ^{i}, θ^{j}} = \sum_{a} (δ_{a}^{i} δ_{m + a}^{j} - δ_{m + a}^{i} δ_{a}^{j}) = J^{i j}$ ; in the Cartesian chart $J$ is the standard matrix, so ${x^{a}, x^{b}} = {p_{a}, p_{b}} = 0$ and ${x^{a}, p_{b}} = δ_{b}^{a}$ ; this is clause 5.

2.1step 1.1F1F3algebra

Leibniz: by [F1] and the derivation property of $d$ of [F3], ${f, g h} = - d (g h) (X_{f}) = - g d h (X_{f}) - h d g (X_{f}) = g {f, h} + h {f, g}$ , and the second identity follows from skew-symmetry of step 1.1 applied to the first; this proves clause 2.

2.2step 1.2F2F3algebra

Jacobi: substituting the Hessian lemma of step 1.2 into the cyclic sum and using ${f, g} = \nabla f^{T} J \nabla g$ gives ${f, {g, h}} + {g, {h, f}} + {h, {f, g}} = (\nabla f)^{T} J H e s s (g) J \nabla h - (\nabla f)^{T} J H e s s (h) J \nabla g + (\nabla g)^{T} J H e s s (h) J \nabla f - (\nabla g)^{T} J H e s s (f) J \nabla h + (\nabla h)^{T} J H e s s (f) J \nabla g - (\nabla h)^{T} J H e s s (g) J \nabla f$ ; each scalar term is its own transpose, and transposing $(\nabla f)^{T} J H e s s (g) J \nabla h$ gives $(\nabla h)^{T} J^{T} H e s s (g)^{T} J^{T} \nabla f = (\nabla h)^{T} J H e s s (g) J \nabla f$ using the symmetry of the Hessian and $J^{T} = - J$ [F2, F3]; hence the two $H e s s (g)$ -terms differ by a sign and cancel, and the same cancellation applies to the $H e s s (h)$ - and $H e s s (f)$ -terms, so the cyclic sum vanishes; this is clause 3.

3.1step 1.1step 2.1step 2.2step 1.3F2∎

Collection: clause 1 is step 1.1, clause 2 is step 2.1, clause 3 is step 2.2, clause 4 is the coordinate formula of [F2] together with step 1.1, and clause 5 is step 1.3; these identities hold at the stated finite regularities with nested brackets read through the $C^{1}$ coordinate extension. For smooth inputs every bracket and product remains smooth, so the identities supply a Poisson algebra on $C^{\infty} (W_{0})$ .

Remarks

Regularity ledger. Bilinearity, skew-symmetry, the Leibniz rule and the identity ${f, g} = X_{g} (f)$ use only $C^{2}$ data; the Jacobi identity uses only $C^{2}$ functions: a bracket is $C^{1}$ , and its first derivative is the Hessian expression of step 1.2. The pointwise extension to $C^{1}$ slots makes the nested terms meaningful; it does not make the $C^{2}$ class closed. The algebra assertion uses the closed smooth class.
What is not asserted. The bracket is an operation on phase functions of the declared presentation; no statement is made here about the evolution equation $d f / d t = {f, H} + \partial_{t} f$ , about first integrals, or about bracket invariance under canonical maps, which are Evolution of phase functions along the model's Hamiltonian solutions: $d f / d t = \partial_{t} f + {f, H}$ , the first-integral criterion, and the closed bracket algebra of autonomous first integrals and Equal-dimensional canonical transformations of a potential presentation conjugate Hamiltonian fields and transport systems and brackets.

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

Statement

Facts & Assumptions

Proof

Remarks

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