954
Appendix: All Solutions to the Problems
Solution This is a variational problem of the functional with the rotation and time,
finding the various partial derivatives, there are
∂ L
∂ B
= −
1
˜
μ
∇ × E +
∂ B
∂t
+ K
,
∂ L
∂ D
=
1
˜
ε
∇ × H −
∂ D
∂t
− J
∂ L
∂ E
= −
∇ × H −
∂ D
∂t
− J
,
∂ L
∂ H
= ∇ × E +
∂ B
∂t
+ K
∂ L
∂∇×E
= H −
B
˜
μ
,
∂ L
∂∇×H
=
D
˜
ε
− E
The Euler equations corresponding to the Lagrange density function are
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
∇ ×
B
˜
μ
=
∂ D
∂t
+ J
∇ ×
D
˜
ε
= −
∂ B
∂t
− K
∇ × E = −
∂ B
∂t
− K = − ˜
μ
∂ H
∂t
− K
∇ × H =
∂ D
∂t
+ J = ˜
ε
∂ E
∂t
+ J
(1)
The natural boundary conditions are
H −
B
˜
μ
× n
S
= 0,
E −
D
˜
ε
× n
S
= 0
(2)
10.40 Find the Euler equations and boundary conditions expressed by A, the
functional is
Π =
V
1
2
(∇ × A
∗
· v∇ × A + ∇ × A · v∇ × A
∗
) + A
∗
· σ
∂ A
∂t
− A · σ
∂ A
∗
∂t
dV
+
V
(σ A
∗
· ∇φ + σ A · ∇φ
∗
)dV +
1
2
V
(∇φ
∗
· σ ∇Φ + ∇φ · σ ∇Φ
∗
)dV
−
V
( J
∗
s · A + J s · A
∗
)dV −
S
(H t · A
∗
+ H
∗
t · A)dS
Solution Putting
F =
1
2
(∇ × A
∗
· v∇ × A + ∇ × A · v∇ × A
∗
) + A
∗
· σ
∂ A
∂t
− A · σ
∂ A
∗
∂t
+ σ A
∗
· ∇φ + σ A · ∇φ
∗
+
1
2
(∇φ
∗
· σ ∇Φ + ∇φ · σ ∇Φ
∗
) − J
∗
s · A − J s · A
∗
Finding the various partial derivatives
∂ F
∂∇ × A
∗ = v∇ × A,
∂ F
∂ A
∗ =
1
2
σ
∂ A
∂t
+ σ ∇φ − J s
The Euler equation of the functional is