Manuscript XII-B@1 · XII-B.tex · section · 229bc869

A local Euler convention

Density weighting is not source weighting › A local Euler convention · lines 479–506

XII-B@1#eq:Euler

Let $ψ^{a}$ denote finitely many even field components, and let $ℓ (x, j^{r} ψ)$ be a local density of finite jet order. For a multi-index $I$ , write

ψ_{I}^{a} = \partial^{I} ψ^{a}, D_{I} = D_{1}^{I_{1}} \dots D_{4}^{I_{4}} .

Total derivatives act on local jet expressions; when applied to a function of $x$ alone, they are ordinary derivatives.

Compact variations define the Euler expression

EL_{a} (ℓ) = ∣ I ∣ \leq r \sum (- 1)^{∣ I ∣} D_{I} (\frac{\partial ℓ}{\partial ψ _{I} ^{a}}) .

Thus

δ \int ℓ d^{4} x = \int EL_{a} (ℓ) δ ψ^{a} d^{4} x

for compact variations. This is the ordinary local variational calculus; see also KO. No global convergence of an unlocalized action on arbitrary configurations is needed.

Displayed formulas in this passage

XII-B@1#eq:Euler · line 492
$EL_{a} (ℓ) = ∣ I ∣ \leq r \sum (- 1)^{∣ I ∣} D_{I} (\frac{\partial ℓ}{\partial ψ _{I} ^{a}}) .$
unlabeled · line 484
$ψ_{I}^{a} = \partial^{I} ψ^{a}, D_{I} = D_{1}^{I_{1}} \dots D_{4}^{I_{4}} .$
unlabeled · line 499
$δ \int ℓ d^{4} x = \int EL_{a} (ℓ) δ ψ^{a} d^{4} x$

A local Euler convention

Displayed formulas in this passage

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