3.1 Extremal Curve Fields
203
Fig. 3.3 c sin x curves
x
y
0
π
δ
x = a
x = a1
If an inherent curve field or a central curve field is formed by the family of
extremal curves of a variational problem of a functional, then the curve field is called
the extremal curve field or central field of extremal curves. If an extremal curve is
located in the family of extremal curves to form an inherent field (or a center field),
then the extremal curve is called the included in extremal curve field or included
in central field of extremal curve.
The inherent curve field, central curve field and extremal curve field can be referred
to as curve field or field for short.
The above concepts about the field can be extended to the situation of high
dimensional space.
Example 3.1.2 Find the central field of extremal curve of the functional J [y] =
x 1
0 (y
− y
2
)dx containing the extremal curve y = 0, the boundary conditions are
y(0) = 0, y(x 1 ) = y 1 (x 1 < π).
Solution From the Euler equation y
+ y = 0 of the functional, solve for y =
c 1 sin x + c 2 cos x, according to the boundary conditions y(0) = 0, we get c 2 = 0.
So y = c 1 sin x is a family of extremal curves, it forms a central curve field in
0 ≤ x ≤ x 1 (x 1 < π), the center is the coordinate origin (0, 0). From the boundary
condition y(x 1 ) = y 1 , we get c 1 =
y 1
sin x 1
, when y 1 = 0, there are c 1 = 0, y = 0,
it contains in the center field formed by the family of extremal curves y = c 1 sin x
that the center is located at the coordinate origin (0, 0). If x 1 ≥ π in this question,
then there is not the extremal curve field including extremal curve y = 0 taking the
coordinate origin (0, 0) as the center.
203
Fig. 3.3 c sin x curves
x
y
0
π
δ
x = a
x = a1
If an inherent curve field or a central curve field is formed by the family of
extremal curves of a variational problem of a functional, then the curve field is called
the extremal curve field or central field of extremal curves. If an extremal curve is
located in the family of extremal curves to form an inherent field (or a center field),
then the extremal curve is called the included in extremal curve field or included
in central field of extremal curve.
The inherent curve field, central curve field and extremal curve field can be referred
to as curve field or field for short.
The above concepts about the field can be extended to the situation of high
dimensional space.
Example 3.1.2 Find the central field of extremal curve of the functional J [y] =
x 1
0 (y
− y
2
)dx containing the extremal curve y = 0, the boundary conditions are
y(0) = 0, y(x 1 ) = y 1 (x 1 < π).
Solution From the Euler equation y
+ y = 0 of the functional, solve for y =
c 1 sin x + c 2 cos x, according to the boundary conditions y(0) = 0, we get c 2 = 0.
So y = c 1 sin x is a family of extremal curves, it forms a central curve field in
0 ≤ x ≤ x 1 (x 1 < π), the center is the coordinate origin (0, 0). From the boundary
condition y(x 1 ) = y 1 , we get c 1 =
y 1
sin x 1
, when y 1 = 0, there are c 1 = 0, y = 0,
it contains in the center field formed by the family of extremal curves y = c 1 sin x
that the center is located at the coordinate origin (0, 0). If x 1 ≥ π in this question,
then there is not the extremal curve field including extremal curve y = 0 taking the
coordinate origin (0, 0) as the center.