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Table 22.30 Variational calculus: δ dS l 2
tr C l G l = 0 (first part)
E 2 A 1 ,A 2 : Surface element
dS l = det G l dLdB = N cos BdLdB
(22.191)
A 1 1
E 2
dS l =
cos BdLdB
(22.192)
1
E 2 sin 2 B 2
Hilbert invariant
1
G 11
1
G 22
1
G 11
1
G 22
tr C l G l =
x L +
x 2 B +
y L +
y B
(22.193)
1
N 2 cos 2 B x L +
1
M 2
1
N 2 cos 2 B y L +
1
M 2
tr C l G 1
l
x 2 B +
y B
=
(22.194)
integral functional
G 11
x 2 B + dS l 2
G 11
y B
dS l tr C l G 1
l
= dS l 2
J := 2
x L + G 22
y L + G 22
(22.195)
=
|
+
||
1
G 11
x 2 B = dS l 1
:= dS l 1
2
1
G 22
2
x L +
|
2
Grad x
(22.196)
:= dS l 1
2
1
G 11
y B = dS l 1
1
G 22
2
y L +
||
2
Grad y
(22.197)
E A 1 ,A 2 : Cartan 2
leg
∂L = ∂X
C 1 = ∂X
1
G 11
∂X
∂L ÷
(22.198)
∂L
∂B = ∂X
C 2 = ∂X
1
G 22
∂X
∂B ÷
(22.199)
∂B
1
G 11
∂X
∂L + C 2
1
G 22
∂X
∂B
Grad x = C 1
(22.200)
∂y
∂L + C 2
∂y
∂B
1
G 11
1
G 22
Grad y = C 1
(22.201)
1
G 11
1
G 22
2 =
x L +
x 2 B
Grad x
Grad x
|
Grad x
=
(22.202)
1
G 11
1
G 22
2 =
y L +
y B
Grad y
Grad y
|
Grad y
=
(22.203)
“1st variation”
δ|
= 0
(22.204)
δ
||
= 0
(22.205)
= dS l 1
G 11
x B δx B = 0
= dS l
1
G 22
δ
|
Grad x
|
Grad x
x L δx L +
(22.206)
= dS l
= dS l 1
G 11
y B δy B = 0
1
G 22
δ
||
Grad y
|
Grad y
y L δy L +
(22.207)
 
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