344
5 Variational Problems of Conditional Extrema
y
C
A
B
x
O
Fig. 5.3 The relational graph between the ellipse and triangles
extremal curve of integral a. Similarly, if λ 1 = 0, then H is the same as ϕ, the
conditional extremal curve of integral a is the unconditional extreml curve.
Example 5.3.6 Prove: In the triangles that the base side and the area are determined,
the circumference of the isosceles triangle is the shortest. In the triangles that the
base side and the circumference are determined, the area of the isosceles triangle is
the largest.
Proof Drawing an ellipse, such that the base side AB of the triangle is exactly the
length between the two focuses of the ellipse, as shown in Fig. 5.3. According to
the property of the ellipse, the circumferences of the various triangles are equal,
but comparing with the scalene triangle, because the height of the isosceles triangle
ABC is the largest, its area is also the largest, At the moment, the vertex of the
isosceles triangle is at the intersection point of the ellipse and short axis. According
to the principle of reciprocity, for the various triangles that the base side and the
area are determined, the isosceles triangle has the shortest circumference. Quod erat
demonstrandum.
5.4 Extremal Problems of Mixed Type Functionals
In the practical problems, such a functional is frequently encountered, besides
containing usual integral type functional, it also contains additional terms, where
the form of the additional term is different from the form of the usual integral type
functional, This form of the functional is called the functional of mixed type, mixed
type functional or mixed functional, is also called the generalized functional.
5 Variational Problems of Conditional Extrema
y
C
A
B
x
O
Fig. 5.3 The relational graph between the ellipse and triangles
extremal curve of integral a. Similarly, if λ 1 = 0, then H is the same as ϕ, the
conditional extremal curve of integral a is the unconditional extreml curve.
Example 5.3.6 Prove: In the triangles that the base side and the area are determined,
the circumference of the isosceles triangle is the shortest. In the triangles that the
base side and the circumference are determined, the area of the isosceles triangle is
the largest.
Proof Drawing an ellipse, such that the base side AB of the triangle is exactly the
length between the two focuses of the ellipse, as shown in Fig. 5.3. According to
the property of the ellipse, the circumferences of the various triangles are equal,
but comparing with the scalene triangle, because the height of the isosceles triangle
ABC is the largest, its area is also the largest, At the moment, the vertex of the
isosceles triangle is at the intersection point of the ellipse and short axis. According
to the principle of reciprocity, for the various triangles that the base side and the
area are determined, the isosceles triangle has the shortest circumference. Quod erat
demonstrandum.
5.4 Extremal Problems of Mixed Type Functionals
In the practical problems, such a functional is frequently encountered, besides
containing usual integral type functional, it also contains additional terms, where
the form of the additional term is different from the form of the usual integral type
functional, This form of the functional is called the functional of mixed type, mixed
type functional or mixed functional, is also called the generalized functional.