Manuscript XX-A@1 · XX-A.tex · section · 3a586110

Rod equation and spatial boundary condition

Original slice-harmonic rods › Rod equation and spatial boundary condition · lines 478–535

XX-A@1#eq:B1, XX-A@1#eq:B2, XX-A@1#eq:newton, XX-A@1#eq:Y1, XX-A@1#eq:Y2

This is a slice-elliptic prescription, not a hyperbolic coordinate Cauchy problem. There are no rod position and velocity data on an initial time surface to replace its spatial boundary condition.

Normalize its density by the background spatial scale:

B_{s}^{ij} = a (τ)^{- 1} det γ_{s} γ_{s}^{ij} = δ^{ij} + s B_{1}^{ij} + \frac{s ^{2}}{2} B_{2}^{ij} .

Let

C = a^{- 2} γ_{1}, D = a^{- 2} γ_{2},

with spatial indices raised by $δ$ . Then

B_{1} B_{2} = \frac{1}{2} (tr C) I - C, = (\frac{1}{2} tr D + \frac{1}{4} (tr C)^{2} - \frac{1}{2} tr (C^{2})) I - D - (tr C) C + 2 C^{2} .

These formulas follow by multiplying the second-order expansions of the determinant square root and matrix inverse.

Use the Newton inverse

(G f) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{f ( y )}{∣ x - y ∣} d^{3} y, Δ G f = f .

For

R_{s}^{a} = x^{a} + s Y_{1}^{a} + \frac{s ^{2}}{2} Y_{2}^{a},

the complete coefficient equations and solutions are

Δ Y_{1}^{a} Δ Y_{2}^{a} = - \partial_{i} B_{1}^{ia}, = - \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}), Y_{1}^{a} Y_{2}^{a} = - G \partial_{i} B_{1}^{ia}, = - G \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .

Displayed formulas in this passage

XX-A@1#eq:B1 · line 495
$B_{1} B_{2} = \frac{1}{2} (tr C) I - C, = (\frac{1}{2} tr D + \frac{1}{4} (tr C)^{2} - \frac{1}{2} tr (C^{2})) I - D - (tr C) C + 2 C^{2} .$
XX-A@1#eq:B2 · line 495
$B_{1} B_{2} = \frac{1}{2} (tr C) I - C, = (\frac{1}{2} tr D + \frac{1}{4} (tr C)^{2} - \frac{1}{2} tr (C^{2})) I - D - (tr C) C + 2 C^{2} .$
XX-A@1#eq:newton · line 508
$(G f) (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{f ( y )}{∣ x - y ∣} d^{3} y, Δ G f = f .$
XX-A@1#eq:Y1 · line 521
$Δ Y_{1}^{a} Δ Y_{2}^{a} = - \partial_{i} B_{1}^{ia}, = - \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}), Y_{1}^{a} Y_{2}^{a} = - G \partial_{i} B_{1}^{ia}, = - G \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .$
XX-A@1#eq:Y2 · line 521
$Δ Y_{1}^{a} Δ Y_{2}^{a} = - \partial_{i} B_{1}^{ia}, = - \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}), Y_{1}^{a} Y_{2}^{a} = - G \partial_{i} B_{1}^{ia}, = - G \partial_{i} (B_{2}^{ia} + 2 B_{1}^{ij} \partial_{j} Y_{1}^{a}) .$
unlabeled · line 485
$B_{s}^{ij} = a (τ)^{- 1} det γ_{s} γ_{s}^{ij} = δ^{ij} + s B_{1}^{ij} + \frac{s ^{2}}{2} B_{2}^{ij} .$
unlabeled · line 491
$C = a^{- 2} γ_{1}, D = a^{- 2} γ_{2},$
unlabeled · line 517
$R_{s}^{a} = x^{a} + s Y_{1}^{a} + \frac{s ^{2}}{2} Y_{2}^{a},$

Rod equation and spatial boundary condition

Displayed formulas in this passage

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