Manuscript Psy02-B@2 · Psy02-B-v2.tex · theorem · cfd06e2e

Theorem: Weighted second-jet Euler identity

Track B foundations: density weighting, response, and roots › A fully specified parameterized Euler interface · lines 1502–1555 · proof included or linked

Psy02-B@2#thm:Euler, Psy02-B@2#eq:weightedEuler

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Put

A_{I}^{α} = \frac{\partial ℓ _{3}}{\partial ( \partial _{α} γ _{I} )}, B_{I}^{α β} = \frac{\partial ℓ _{3}}{\partial ( \partial _{α} \partial _{β} γ _{I} )}, B_{I}^{α β} = B_{I}^{β α} .

Then

EL_{I} (σ ℓ_{3}) = σ J_{3, I} + (\partial_{α} σ) (- A_{I}^{α} + 2 \partial_{β} B_{I}^{α β}) + (\partial_{α} \partial_{β} σ) B_{I}^{α β} .

There are no derivatives of $σ$ beyond order two. If the chosen density's coefficient identifications are

B_{I}^{α β} = 2 C_{2, I}^{α β}, - A_{I}^{α} + 2 \partial_{β} B_{I}^{α β} = - 2 M_{2, I}^{α},

then

J_{3, D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .

For a time-only ramp the extra terms are $2 C_{2}^{μν 00} σ_{ε}^{''} - 2 M_{2}^{μν 0} σ_{ε}^{'}$ .

Proof. Apply

EL_{I} = \frac{\partial}{\partial γ _{I}} - \partial_{α} \frac{\partial}{\partial ( \partial _{α} γ _{I} )} + \partial_{α} \partial_{β} \frac{\partial}{\partial ( \partial _{α} \partial _{β} γ _{I} )}

and expand each product derivative. Symmetry of $B^{α β}$ combines the two first-ramp-derivative terms. The coefficient identifications give the remaining formulas.

Displayed formulas in this passage

Psy02-B@2#eq:weightedEuler · line 1513
$EL_{I} (σ ℓ_{3}) = σ J_{3, I} + (\partial_{α} σ) (- A_{I}^{α} + 2 \partial_{β} B_{I}^{α β}) + (\partial_{α} \partial_{β} σ) B_{I}^{α β} .$
unlabeled · line 1505
$A_{I}^{α} = \frac{\partial ℓ _{3}}{\partial ( \partial _{α} γ _{I} )}, B_{I}^{α β} = \frac{\partial ℓ _{3}}{\partial ( \partial _{α} \partial _{β} γ _{I} )}, B_{I}^{α β} = B_{I}^{β α} .$
unlabeled · line 1524
$B_{I}^{α β} = 2 C_{2, I}^{α β}, - A_{I}^{α} + 2 \partial_{β} B_{I}^{α β} = - 2 M_{2, I}^{α},$
unlabeled · line 1530
$J_{3, D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
unlabeled · line 1545
$EL_{I} = \frac{\partial}{\partial γ _{I}} - \partial_{α} \frac{\partial}{\partial ( \partial _{α} γ _{I} )} + \partial_{α} \partial_{β} \frac{\partial}{\partial ( \partial _{α} \partial _{β} γ _{I} )}$

Theorem: Weighted second-jet Euler identity

Displayed formulas in this passage

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