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Closed-system energy, momentum and angular balance conditional on a matter model

Physical scope

The conditional total balances obtained by adding explicitly assumed matter balance equations to the vacuum field balances of the same framework: local zero-source identities for total energy, momentum and orbital angular momentum and their fixed-volume boundary-flux forms. The matter balances are additional assumptions about a specified matter model, not consequences of the Maxwell postulates; no closed system, no conservation without flux conditions and no empirical claim is asserted.

Statement

Assume the hypotheses of Local and fixed-volume electromagnetic energy balance and Local and fixed-volume momentum balance for smooth sources: the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches and clause (ii) of The Lorentz coupling: external force on a test charge and the continuum force and power densities on the laboratory region $U ≅ I \times Ω$ , jointly $C^{1}$ fields of The electric and magnetic fields as fixed-time spatial vector fields, continuous sources, the constants of the framework, and the field objects $u, S, g, Π$ of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum. Suppose in addition that matter fields $u_{m} : Ω \times I \to R$ , $S_{m} : Ω \times I \to R^{3}$ , $g_{m} : Ω \times I \to R^{3}$ and a symmetric matrix-valued map $Π_{m} : Ω \times I \to R^{3 \times 3}$ are of class $C^{1}$ , with $Π_{m, i j} = Π_{m, j i}$ and $(div Π_{m})_{i} = \sum_{j = 1}^{3} \partial_{j} Π_{m, i j}$ . Their SI units are respectively $J m^{- 3}$ , $W m^{- 2}$ , $k g m^{- 2} s^{- 1}$ and $N m^{- 2}$ , matching the field quantities added below. Assume they satisfy the matter balance equations

$\partial_{t} u_{m} + div S_{m} = J \cdot E, \partial_{t} g_{m} + div Π_{m} = f,$

pointwise on $Ω \times I$ , where $f = ρ E + J \times B$ and $J \cdot E$ are the coupling densities of clause (ii). Put $u_{t o t} : = u + u_{m}$ , $S_{t o t} : = S + S_{m}$ , $g_{t o t} : = g + g_{m}$ , $Π_{t o t} : = Π + Π_{m}$ (symmetric) and $ℓ_{t o t} : = x \times g_{t o t}$ . Then:

(a) [Local conservation] On $Ω \times I$ ,

$\partial_{t} u_{t o t} + div S_{t o t} = 0, \partial_{t} g_{t o t} + div Π_{t o t} = 0, \partial_{t} ℓ_{t o t} + div (x \times Π_{t o t}) = 0$

componentwise in the last two identities.

(b) [Fixed-volume conservation] Let $Ω^{'} \subseteq Ω$ be open with $\overline{Ω^{'}} \subseteq Ω$ carrying the finite piecewise $C^{1}$ presentation of Divergence for finite piecewise C1 presentations, let $t_{0} < t_{1}$ with $(t_{0}, t_{1}) \subseteq I$ , assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), and let $n$ be the outward unit normal. Then for every $t \in (t_{0}, t_{1})$ ,

$\frac{d}{d t} \int_{Ω^{'}} u_{t o t} d x = - \oint_{\partial Ω^{'}} S_{t o t} \cdot n d S, \frac{d}{d t} \int_{Ω^{'}} g_{t o t, i} d x = - \oint_{\partial Ω^{'}} (\sum_{j = 1}^{3} Π_{t o t, i j} n_{j}) d S, \frac{d}{d t} \int_{Ω^{'}} ℓ_{t o t, i} d x = - \oint_{\partial Ω^{'}} (\sum_{j = 1}^{3} (x \times Π_{t o t})_{i j} n_{j}) d S,$

the closed-surface integrals being finite face sums of that divergence theorem. Consequently each of the three totals is constant on $(t_{0}, t_{1})$ if and only if the corresponding boundary flux integral vanishes for every $t \in (t_{0}, t_{1})$ .

(c) [Scope] The matter balances are additional, explicitly stated assumptions, not consequences of the Maxwell postulates: a prescribed current $(ρ, J)$ does not by itself supply matter fields satisfying them, and "closed system" is a modelling choice about which boundary terms are included. The hypotheses and exclusions of the two field theorems are inherited, and no empirical claim is made.

Facts & Assumptions

Given: The hypotheses of Local and fixed-volume electromagnetic energy balance and Local and fixed-volume momentum balance for smooth sources; the field objects of Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum; and the $C^{1}$ matter fields $u_{m}, S_{m}, g_{m}$ and symmetric $C^{1}$ $Π_{m}$ satisfying the two matter balance equations pointwise. In clause (b) also the admissible $Ω^{'}$ , the interval $(t_{0}, t_{1})$ and ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Field balances: $\partial_{t} u + div S = - J \cdot E$ and $\partial_{t} g + div Π = - f$ pointwise on $Ω \times I$ , with $u, S, g, Π$ of class $C^{1}$ , $Π$ symmetric, and $f$ continuous (Local and fixed-volume electromagnetic energy balance, Local and fixed-volume momentum balance for smooth sources).

[F2]

Angular field balance: with $ℓ = x \times g$ and $x \times Π$ as in Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum, $\partial_{t} ℓ + div (x \times Π) = - x \times f$ pointwise on $Ω \times I$ , under the hypotheses of [F1] (Orbital field angular momentum and torque balance).

[F3]

Matter balances: $\partial_{t} u_{m} + div S_{m} = J \cdot E$ and $\partial_{t} g_{m} + div Π_{m} = f$ pointwise on $Ω \times I$ , with $u_{m}, S_{m}, g_{m}, Π_{m}$ of class $C^{1}$ and $Π_{m}$ symmetric; the coupling densities are those of clause (ii) of The Lorentz coupling: external force on a test charge and the continuum force and power densities.

[F4]

Sum rules: for differentiable real functions of the time variable, sums and scalar multiples have the expected derivatives (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 $g (c) \neq 0$ ); and the divergence is linear in its argument, being a finite sum of coordinate partial derivatives (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F5]

Fixed-volume tools: compact-parameter differentiation (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface); the divergence theorem on the finite piecewise $C^{1}$ presentation with faces counted once off the edge set, under ${A C}_{ω}$ (Divergence for finite piecewise C1 presentations); the Lebesgue-nullity of its boundary by the graph-coordinate Fubini and rigid change of variables in Proof step 4.1 of that divergence theorem, followed by finite subadditivity over its faces; the vanishing of nonnegative measurable integrals over null sets (A nonnegative integral over a null set vanishes); and the fact that a continuous function on an interval whose derivative vanishes at every interior point is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[F6]

The quadratic stress lemma in the form used for the total angular identity: if $Π$ is $C^{1}$ and symmetric, $g$ continuous, and $\partial_{t} g + div Π = - F$ pointwise for continuous $F$ , then $\partial_{t} (x \times g) + div (x \times Π) = - x \times F$ (The quadratic stress identity and the local angular-balance identity).

[F7]

Identities of the cross product and its components used in the total flux tensor: $(x \times Π_{t o t})_{i j} = \sum_{k, l = 1}^{3} ϵ_{i k l} x_{k} Π_{t o t, l j}$ and $Π_{t o t}$ symmetric (Momentum density, Maxwell stress, momentum flux and orbital angular density in vacuum, The cross product in $R^{3}$ ).

Proof

technique · direct; add each matter balance to the corresponding field balance to obtain the total energy and momentum identities, obtain the total angular identity by adding the field angular balance to the angular identity implied by the matter momentum balance, and integrate with the divergence theorem

1.1givenF1F3F4algebra

Total energy and momentum: adding the two identities of [F1] to the two matter balances of [F3] and using the sum rules [F4], $\partial_{t} u_{t o t} + div S_{t o t} = (\partial_{t} u + div S + J \cdot E) + (\partial_{t} u_{m} + div S_{m} - J \cdot E) = 0$ and $\partial_{t} g_{t o t} + div Π_{t o t} = (\partial_{t} g + div Π + f) + (\partial_{t} g_{m} + div Π_{m} - f) = 0$ pointwise on $Ω \times I$ , since $Π_{t o t} = Π + Π_{m}$ and the coupling terms cancel between field and matter.

1.2givenF2F3F4F6F7algebra

Total angular: the field angular identity [F2] reads $\partial_{t} (x \times g) + div (x \times Π) = - x \times f$ ; the matter momentum balance $\partial_{t} g_{m} + div Π_{m} = f$ of [F3], written as $\partial_{t} g_{m} + div Π_{m} = - (- f)$ with the continuous $F : = - f$ , enters the lemma [F6] and gives $\partial_{t} (x \times g_{m}) + div (x \times Π_{m}) = - x \times (- f) = x \times f$ , where the tensor $x \times Π_{m}$ is the one defined inside [F6] for the symmetric $Π_{m}$ . Adding the two with [F4] yields $\partial_{t} ℓ_{t o t} + div (x \times Π_{t o t}) = 0$ pointwise, using $ℓ_{t o t} = x \times (g + g_{m})$ and $Π_{t o t} = Π + Π_{m}$ as in [F7].

1.3givenF1F3F5algebra

Fixed-volume setup: let $Ω^{'}$ , $t_{0} < t_{1}$ be as in clause (b), fix $t \in (t_{0}, t_{1})$ and choose $a, b$ with $t_{0} < a < t < b < t_{1}$ . The closures are compact and contained in the open $Ω$ ; all six quantities $u_{t o t}, S_{t o t}, g_{t o t, i}, Π_{t o t, i j}, ℓ_{t o t, i}, (x \times Π_{t o t})_{i j}$ and their time derivatives are continuous on a neighbourhood of $\overline{Ω^{'}} \times [a, b]$ by [F1], [F3] and the sum rules; the boundary is Lebesgue-null by [F5], so integrals over $Ω^{'}$ and its closure agree, and [F5] applies to each of the three total quantities separately.

2.1step 1.1step 1.2step 1.3F5algebra

Fixed-volume identities: by steps 1.1-1.3, $\frac{d}{d t} \int_{Ω^{'}} u_{t o t} d x = \int_{Ω^{'}} \partial_{t} u_{t o t} d x = - \int_{Ω^{'}} div S_{t o t} d x = - \oint_{\partial Ω^{'}} S_{t o t} \cdot n d S$ , and likewise for $g_{t o t}$ with the vector fields $(Π_{t o t, i 1}, Π_{t o t, i 2}, Π_{t o t, i 3}) (\cdot, t)$ and for $ℓ_{t o t}$ with $((x \times Π_{t o t})_{i 1}, (x \times Π_{t o t})_{i 2}, (x \times Π_{t o t})_{i 3}) (\cdot, t)$ , each $C^{1}$ on a neighbourhood of $\overline{Ω^{'}}$ and each integrated with the divergence theorem of [F5]; this proves the three displayed identities and their finite face sums. If a total function has vanishing derivative at every point of $(t_{0}, t_{1})$ then it is constant there by [F5], and if it is constant then its derivative vanishes and the displayed identity forces the corresponding flux to vanish; this is the asserted equivalence for each of the three totals.

3.1step 2.1F3F5∎

Clause (c): the matter balances were assumed, not derived; the coupling densities enter exactly through clause (ii) of [F3], and the boundary terms were retained, so constancy on the stated interval is equivalent to vanishing of its net boundary flux throughout that interval; a nonzero flux describes exchange to be tracked in the chosen system; the exclusions of the field theorems remain in force and no empirical content is added.

Remarks

The cancellation is formal, the assumptions are physical. The proof only adds two pointwise identities; the physical content lies in the hypothesis that a matter model exists whose balances carry exactly the coupling densities $J \cdot E$ and $f$ with the opposite signs. A prescribed current does not supply such matter fields.
Totality is a modelling choice. "Closed system" here means that the chosen volume's boundary flux is either neglected because it vanishes or tracked separately; the theorem asserts the exact boundary-flux forms and nothing more.

Closed-system energy, momentum and angular balance conditional on a matter model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources