10.9 Introduction to the Famous Scientists
745
particle; x is distance from the interface into the substrate. Find the Euler
equation and natural boundary condition of the functional.
10.19 The functional of the time-dependent Schrödinger equation is
J [ϕ] =
V
[(V ϕ + 2iϕ 0 − iϕ)ϕ + k∇ϕ · ∇ϕ]dV
Find its Euler equation.
10.20 Find the Ostrogradsky equation of the functional J [u]
=
1
m
V (2 f u + |∇u|
2
)dV
n
, where, n ≥ 1.
10.21 Find the Euler equations and corresponding boundary conditions of the
functional J [u] =
1
2
V
1
μ
|∇u|
2
+ iωσ u
2
− 2 ju
dV .
10.22 Find the Euler equations of the functional J [u, v]
=
V (|∇u|
2
+ |∇v|
2
+ αu
2
+ βv
2
− γ u
2 v)dV .
10.23 There is the Lagrange function in the analysis of a complex sound intensity
field
L =
G 0
ρc 2 e
ψ
(∇ϕ)
2
k 2 +
(∇ψ)
2
4k 2 − 1
where, G 0 is the auto-spectra of sound pressure; ρ is the density of the
medium; c is the sound velocity; k is wave number, k = ω/c, here ω is
the circle frequency. Write out its Euler equations and natural boundary
conditions.
10.24 In the process of adiabatic compression heating for the plasma of magnetic
confinement, the Lagrange function is
L =
(∇ψ) 2
2R 2
1 −
F 2
ρ
−
1
2
ρ
ρ − F 2
I
R
+ R FΩ
2
−
ρ R 2 Ω 2
2
+
ρ γ S
γ − 1
− ρ H
R
where, F, I, Ω and S are all arbitrary functions of ψ, find the Euler equation
and natural boundary condition of the functional.
10.25 The non-Darcy flow problem. The functional is J
=
A K (|∇ϕ|)(ϕ
2
x + ϕ
2
y )dx dy, where, ϕ is the piezometric head; K (|∇ϕ|) is
the water conductance. Find its Euler equation.
10.26 Find the Euler equation of the functional
J [u]
=
V
|∇u| ln(1 + |∇u|) + λ 1
f
u
+ λ 2
f
u
+ a
2
dV .
10.27 Find the Euler equation of the functional J [u] =
V
|∇u − a|
2 dV .
10.28 Find the Euler equation of the functional
J [ f ]
=
V
1
2
f
2
+
b
2
6
∇ f · ∇g + ug f
dV .