362
5 Variational Problems of Conditional Extrema
5.11 Find the Euler equation of the minimum of the functional J [y, z] =
1
√
2g
1
0
1+y z
x
d x, the constraint condition is y = z + 1, the endpoint
conditions are y(0) = 0, y(1) = b.
5.12 In information theory, when researching the information quantity problem of
the information source, the different information source variable x represents
the different information. If the information source variable x changes in the
interval [−a, a], the probability distribution density of the information is the
continuous function p(x), find the best probability distribution density p(x),
such that the information entropy
J [ p(x)] = −
a
−a
p(x) ln[kp(x)]dx
obtains the maximum, the additional condition is
a
−a p(x)dx = 1, where, k is
a constant.
5.13 Using spherical coordinates (r, ϕ, θ), find the geodesic line from point
A(R, 0, 0) to point B
0,
√
2
2
R,
√
2
2
R
on the sphere x
2
+ y
2
+ z
2
= R
2 .
5.14 Find the extremal curve of the functional J =
1
2
x 1
x 0
(y
2
+ u
2
)dx, the boundary
conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 , the constraint condition is y
= u − y.
5.15 Find the extremal curve of the isoperimetric problem, the functional is J [y] =
1
0 y
2 dx, the boundary conditions are y(0) = 0, y(1) =
1
4
, which follows the
isoperimetric conditions
1
0 (y − y
2
)dx =
1
12
.
5.16 Find the solution of the minimum problem of the functional J [y] =
1
−1 (y
2
− k
2 y
2
)dx, the boundary conditions are y(−1) = y(1) = 0, the
additional condition is
1
−1 y
2 dx = 1.
5.17 Let the known function f (x, y) ∈ C(D), σ (Γ ), p(Γ ) ∈ C(Γ ), Γ is the
boundary of D. Write the Euler equation and the natural boundary condition
of the mixed type functional
J [u] =
¨
D
[(u
2
x + u
2
y ) + 2 f (x, y)u]d xd y +
Γ
[σ (Γ )u
2
+ 2 p(Γ )u]d Γ
5.18 Let the known function p(x, y) ∈ C
1
(D), q(x, y), f (x, y) ∈ C(D),
σ (Γ ) ∈ C(Γ ), Γ is the boundary of D. Write the Euler equation and the
natural boundary condition of the mixed type functional
J [u] =
¨
D
[ p(x, y)(u 2
x + u 2
y ) + q(x, y)u 2 − 2 f (x, y)u]d xd y +
Γ
σ (Γ )u 2 d Γ
5.19 Find the Euler equation and the corresponding boundary condition of
the acoustic field functional J =
1
2
˝
V (|∇ p|
2
− k
2 p
2
− 2iωρqp)d V +
1
2
˜
S
iωρ
Z
p
2 d S.
5 Variational Problems of Conditional Extrema
5.11 Find the Euler equation of the minimum of the functional J [y, z] =
1
√
2g
1
0
1+y z
x
d x, the constraint condition is y = z + 1, the endpoint
conditions are y(0) = 0, y(1) = b.
5.12 In information theory, when researching the information quantity problem of
the information source, the different information source variable x represents
the different information. If the information source variable x changes in the
interval [−a, a], the probability distribution density of the information is the
continuous function p(x), find the best probability distribution density p(x),
such that the information entropy
J [ p(x)] = −
a
−a
p(x) ln[kp(x)]dx
obtains the maximum, the additional condition is
a
−a p(x)dx = 1, where, k is
a constant.
5.13 Using spherical coordinates (r, ϕ, θ), find the geodesic line from point
A(R, 0, 0) to point B
0,
√
2
2
R,
√
2
2
R
on the sphere x
2
+ y
2
+ z
2
= R
2 .
5.14 Find the extremal curve of the functional J =
1
2
x 1
x 0
(y
2
+ u
2
)dx, the boundary
conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 , the constraint condition is y
= u − y.
5.15 Find the extremal curve of the isoperimetric problem, the functional is J [y] =
1
0 y
2 dx, the boundary conditions are y(0) = 0, y(1) =
1
4
, which follows the
isoperimetric conditions
1
0 (y − y
2
)dx =
1
12
.
5.16 Find the solution of the minimum problem of the functional J [y] =
1
−1 (y
2
− k
2 y
2
)dx, the boundary conditions are y(−1) = y(1) = 0, the
additional condition is
1
−1 y
2 dx = 1.
5.17 Let the known function f (x, y) ∈ C(D), σ (Γ ), p(Γ ) ∈ C(Γ ), Γ is the
boundary of D. Write the Euler equation and the natural boundary condition
of the mixed type functional
J [u] =
¨
D
[(u
2
x + u
2
y ) + 2 f (x, y)u]d xd y +
Γ
[σ (Γ )u
2
+ 2 p(Γ )u]d Γ
5.18 Let the known function p(x, y) ∈ C
1
(D), q(x, y), f (x, y) ∈ C(D),
σ (Γ ) ∈ C(Γ ), Γ is the boundary of D. Write the Euler equation and the
natural boundary condition of the mixed type functional
J [u] =
¨
D
[ p(x, y)(u 2
x + u 2
y ) + q(x, y)u 2 − 2 f (x, y)u]d xd y +
Γ
σ (Γ )u 2 d Γ
5.19 Find the Euler equation and the corresponding boundary condition of
the acoustic field functional J =
1
2
˝
V (|∇ p|
2
− k
2 p
2
− 2iωρqp)d V +
1
2
˜
S
iωρ
Z
p
2 d S.