5.3 Isoperimetric Problems
333
Taking y 1 , y 2 , …, y n , y
1 , y
2 , …, y
n , z
1 , z
2 , …, z
m , λ i (x) as the independent
functions, the Euler equations given by the variation of the functional (5.3.9) are
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
∂ H
∂ y j
−
d
dx
∂ H
∂ y
j
= 0 ( j = 1, 2, . . . , n)
∂ H
∂z i
−
d
dx
∂ H
∂z
i
= 0 (i = 1, 2, . . . , m)
ϕ i − z
i (x) = 0
(i = 1, 2, . . . , m)
(5.3.11)
Substituting the expression (5.3.10) into Eq. (5.3.11), we get
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ F
∂ y j
+
m
i=1
λ i (x)
∂ϕ i
∂ y j
−
d
dx
∂ F
∂ y
j
+
m
i=1
λ i (x)
∂ϕ i
∂ y j
= 0 ( j = 1, 2, . . . , n)
d
dx
λ i (x) = 0 (i = 1, 2, . . . , m)
ϕ i − z
i (x) = 0 (i = 1, 2, . . . , m)
(5.3.12)
It is observed from the second equation of Eq. (5.3.12) that λ i (x) are the constants,
therefore λ i (x) should be written as λ i . The first equations of Eq. (5.3.12) are the
same as the Euler Eq. (5.3.5) or (5.3.6) of the functional (5.3.4), where
F = F(x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n )
ϕ i = ϕ i (x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n ) (i = 1, 2, . . . , m)
λ i are the constants, thus the theorem has been proved. Quod erat demonstrandum.
Generally, under the constraint condition that a group of values of the functional
are given, finding the extremal problem of the functional is called the generalized
isoperimetric problem. While the isoperimetric problem mentioned in Chap. 2 is
called the special isoperimetric problem. In 1870, Weierstrass in a speech perfectly
solved the isoperimetric problem of the general two-dimensional domain by means
of variational methods, soon after, Schwarz gave the strict proof of isoperimetric
problem of three-dimensional domain.
Example 5.3.1 The isoperimetric problem. In all curves joining two fixed points A
and B with a constant L (L > AB), find out a curve, such that the area surrounded by
it and the line segment AB is the largest. This problem is one of the oldest variational
problem, essentially is also an isoperimetric problem. About this problem there is
also a myth and legend, the ancient Phoenicia had a princess named Dido, she was
the daughter of the King of Tyrus Mutto, married with his uncle, Hercules’ priest
Acerbas (or Acherbas, also known as Sychaeus or Sichaeus) for his wife. After
Dido’s father died, her brother Pygmalion ascended the throne, in order to encroach
333
Taking y 1 , y 2 , …, y n , y
1 , y
2 , …, y
n , z
1 , z
2 , …, z
m , λ i (x) as the independent
functions, the Euler equations given by the variation of the functional (5.3.9) are
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
∂ H
∂ y j
−
d
dx
∂ H
∂ y
j
= 0 ( j = 1, 2, . . . , n)
∂ H
∂z i
−
d
dx
∂ H
∂z
i
= 0 (i = 1, 2, . . . , m)
ϕ i − z
i (x) = 0
(i = 1, 2, . . . , m)
(5.3.11)
Substituting the expression (5.3.10) into Eq. (5.3.11), we get
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂ F
∂ y j
+
m
i=1
λ i (x)
∂ϕ i
∂ y j
−
d
dx
∂ F
∂ y
j
+
m
i=1
λ i (x)
∂ϕ i
∂ y j
= 0 ( j = 1, 2, . . . , n)
d
dx
λ i (x) = 0 (i = 1, 2, . . . , m)
ϕ i − z
i (x) = 0 (i = 1, 2, . . . , m)
(5.3.12)
It is observed from the second equation of Eq. (5.3.12) that λ i (x) are the constants,
therefore λ i (x) should be written as λ i . The first equations of Eq. (5.3.12) are the
same as the Euler Eq. (5.3.5) or (5.3.6) of the functional (5.3.4), where
F = F(x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n )
ϕ i = ϕ i (x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n ) (i = 1, 2, . . . , m)
λ i are the constants, thus the theorem has been proved. Quod erat demonstrandum.
Generally, under the constraint condition that a group of values of the functional
are given, finding the extremal problem of the functional is called the generalized
isoperimetric problem. While the isoperimetric problem mentioned in Chap. 2 is
called the special isoperimetric problem. In 1870, Weierstrass in a speech perfectly
solved the isoperimetric problem of the general two-dimensional domain by means
of variational methods, soon after, Schwarz gave the strict proof of isoperimetric
problem of three-dimensional domain.
Example 5.3.1 The isoperimetric problem. In all curves joining two fixed points A
and B with a constant L (L > AB), find out a curve, such that the area surrounded by
it and the line segment AB is the largest. This problem is one of the oldest variational
problem, essentially is also an isoperimetric problem. About this problem there is
also a myth and legend, the ancient Phoenicia had a princess named Dido, she was
the daughter of the King of Tyrus Mutto, married with his uncle, Hercules’ priest
Acerbas (or Acherbas, also known as Sychaeus or Sichaeus) for his wife. After
Dido’s father died, her brother Pygmalion ascended the throne, in order to encroach