626
10 Variational Problems of Functionals with Vector, Tensor …
J =
t 1
t 0
V
LdV dt =
t 1
t 0
V
1
2
ε E
2
−
1
μ
B
2
− ρϕ + J · A
dV dt
where, there are the expressions of the relation B = ∇ × A, E = −∇ϕ −
∂ A
∂t
=
−∇ϕ − A t , in medium of neither the piezoelectric effect and no the piezomagnetic
effect, there are linear constitutive relations B = μH, D = ε E, J c = σ E, here H
is the magnetic field intensity, and D is the electric displacement vector or electric
flux density, μ is the magnetic permeability of the magnetic medium, μ = μ r μ 0 ,
μ r is the relative magnetic permeability of magnetic medium, μ 0 is the vacuum (or
free space) permeability of magnetic medium, ε is the dielectric constant of matter,
ε = ε r ε 0 , ε r is the relative dielectric constant of material, ε 0 is the vacuum (or
free space) dielectric constant of material, and the velocity of light in vacuum (or
free space) is c =
1
√ μ 0 ε 0
, σ is the electrical conductivity or conductive coefficient.
Find the Euler equations of the functional, and find D’Alembert equation, telegraph
equation and Poynting equation.
Solution According to given the functional relations in the problem, the original
functional can be written as
J =
t 1
t 0
V
1
2
ε E
2
−
1
μ
B
2
− ρϕ + J · A
dV dt
=
t 1
t 0
V
ε
2
(−∇ϕ − A t )
2
−
1
2μ
(∇ × A)
2
− ρϕ + J · A
dV dt
=
t 1
t 0
V
ε
2
(∇ϕ)
2
+ ε∇ϕ · A t +
ε
2
A
2
t −
1
2μ
(∇ × A)
2
− ρϕ + J · A
dV dt
(1)
Calculating the various partial derivatives
∂ L
∂ A
= J,
∂ L
∂∇ × A
= −
∇ × A
μ
= −
B
μ
= −H,
∂ L
∂ A t
= ε∇ϕ + ε A t = −ε E = −D
∂ L
∂ϕ
= −ρ,
∂ L
∂∇ϕ
= ε∇ϕ + ε
∂ A
∂t
= −ε E = −D
The Euler equations of the functional are
J − ∇ × H + D t = 0
( 2 )
∇ · D − ρ = 0
( 3 )
Equation (2) can be written as
∇ × H = J +
∂ D
∂t
(4)