724
10 Variational Problems of Functionals with Vector, Tensor …
Find its Euler equation.
Solution Let the integrand F = u∇ : σ − u · f , find the various partial derivatives
∂ F
∂ u
= − f ,
∂ F
∂ u∇
= σ
The Euler equation of the functional is
∇ · σ + f = 0 (in V )
(2)
According to the boundary conditions, there is
S σ
δu · σ · n dS −
S σ
δu · σ · n dS =
S σ
δu · (σ − σ ) · n dS = 0, that is
(σ − σ )| S σ = 0 (on S σ )
(3)
Example 10.7.3 The effective free energy of the superconductor in the case of the
presence of magnetic field and prestrain can be written as
f = α|ψ|
2 +
1
2
β|ψ|
4 +
1
2m ∗
−i∇ψ − e
∗ Aψ
2 +
1
2μ 0
|∇ × A|
2 +
1
2
C i jkl ε i j ε kl − aθ|ψ|
2 −
1
2
bθ|ψ|
4
(1)
where, ψ is the wave function, ε i j is the strain components, θ is the volume change
in the superconductor, a and b are constants, C i jkl is the elasticity tensor, A is the
magnetic vector potential, m
∗ is the Cooper pair mass, and e
∗ is the charge, e
∗
= 2e.
Find the Euler equations and natural boundary conditions of the functional.
Solution According to the property of the conjugate function, there are |ψ|
2
= ψψ
∗ ,
|−i∇ψ − e
∗ Aψ|
2
= (−i∇ψ − e
∗ Aψ) · (i∇ψ
∗
− e
∗ Aψ
∗
), the Eq. (4.14) can
be rewritten as
−i∇ψ − e
∗ Aψ
2 = (−i∇ψ − e
∗ Aψ) · (i∇ψ
∗
− e
∗ Aψ
∗
), |ψ|
2
= ψψ
∗
Equation (1) is rewritten as
f = (α − aθ)ψψ
∗
+
1
2
(β − bθ)ψ
2
ψ
∗2
+
1
2m ∗ (−i∇ψ − e
∗ Aψ) · (i∇ψ
∗
− e
∗ Aψ
∗
)
+
1
2μ 0
|∇ × A|
2
+
1
2
C i jkl ε i j ε kl
Find the various partial derivatives