Appendix: All Solutions to the Problems
955
∇ × (v∇ × A) + σ
∂ A
∂t
+ σ ∇φ − J s = 0
The boundary condition is
[(v∇ × A) × n − H t ]| S = 0
10.41 In magnetostatics, there is the functional with the Lagrange multiplier λ
J [H, λ] =
V
1
2
(H · ∇ × H) − H · J − λ∇ · μH
dV
where, H is the magnetic field intensity; μ is the magnetic permeability of magnetic
medium, it is a constant. Find the Euler equations of the functional.
Solution Putting F =
1
2
(H · ∇ × H) − H · J − λ∇ · μH, Finding the various
partial derivatives
∂ F
∂ H
=
1
2
∇ × H − J,
∂ F
∂∇ · H
= −λμ,
∂ F
∂∇ × H
=
1
2
H,
∂ F
∂λ
= −∇ · μH
After arrangement, the Euler equations of the functional can be obtained
∇ × H − J + μ∇λ = 0
∇ · (μH) = 0
Taking note that B = μH, here B is the magnetic induction, the second equation
of the above equations represents the Gauss law of the magnetic field, namely the
lines of magnetic induction are always closed.
10.42 In the study of magnetohydrodynamics it will encounter the following
Lagrange density function
L =
1
2μ 0
B
2
−
1
2
k 0 E
2
+
1
2
ρv
2
− ρε − α
dρ
dt
+ ρ∇ · v
− βρ
dS
dt
where, μ 0 and k 0 are constants; α and β are two Lagrange multipliers; ε and S are
the specific internal energy and the specific entropy respectively, and there are
E =
1
μ 0
(∇ × M − P), B = μ 0
∇ψ + P × v +
∂ M
∂t
where, P is the polarization; ψ and M are the scalar potential and the vector potential
respectively in magnetohydrodynamics. Find the corresponding Euler equations.