104
2 Variational Problems with Fixed Boundaries
where, a 1 , a 2 , b 1 and b 2 are all arbitrary constants, then the J [u, v] is called the
symmetric bilinear functional on u and v. If it only satisfies rear two conditions,
the J [u, v] is called the bilinear functional on u and v. For the bilinear functional,
if let u = v, J [u, u] can be obtained, the J [u, u] is called the quadratic functional.
If the functional (2.3.1) is second-order continuous, and its increment can be
expressed as J = L[y(x), δy] + d[y(x), δy], where d[y, δy] is the higher order
infinitesimal of δy, then the functional is called differentiable at y = y(x), and
L[y, δy] is called the first variation of the functional J [y(x)] on y(x), it is also
called the variation of a functional or variation, it is written as δ J [y(x)], δ J [y] or
δ J , that is
δ J =
x 1
x 0
[F y (x, y, y
)δy + F y (x, y, y
)δy
]dx
=
x 1
x 0
(F y δy + F y δy
)dx =
x 1
x 0
(F y εη + F y εη
)dx
= ε
x 1
x 0
(F y η + F y η
)dx
(2.3.15)
The variational of a functional was a concept posed by Lagrange in 1762. The
variational of a functional is a generalization of the concept of the variation of a
function.
The above definition shows that the variation δ J of the functional J has the
following two properties:
(1) The difference between the increment J and variation δ J of a functional and
is higher order infinitesimal than first order distance d 1 (y, y 1 ), the variation δ J
of a functional is the main linear part of the increment J of a functional (i.e.
main linear part);
(2) The integrand of the variation δ J is a linear function on η and η
.
Using the variation δ J of a functional to express the increment J of a functional
is called the linearization of a functional. The linearization of a functional can be
expressed as J = δ J .
Note that in the variation definition of the above functional, the functions y(x)
and y
(x) both are independent variables, which is different from the differential
definition of a function in the higher mathematics in essence. Table 2.1 shows the
corresponding relations of the functional and function.
Example 2.3.1 Verify J [y] =
x 1
x 0
y
2 dx is not a linear functional.
Solution Since
J [cy] =
x 1
x 0
(cy)
2 dx = c
2
x 1
x 0
y
2 dx = c
x 1
x 0
y
2 dx = c J [y]
so J [y] is not a linear functional.
2 Variational Problems with Fixed Boundaries
where, a 1 , a 2 , b 1 and b 2 are all arbitrary constants, then the J [u, v] is called the
symmetric bilinear functional on u and v. If it only satisfies rear two conditions,
the J [u, v] is called the bilinear functional on u and v. For the bilinear functional,
if let u = v, J [u, u] can be obtained, the J [u, u] is called the quadratic functional.
If the functional (2.3.1) is second-order continuous, and its increment can be
expressed as J = L[y(x), δy] + d[y(x), δy], where d[y, δy] is the higher order
infinitesimal of δy, then the functional is called differentiable at y = y(x), and
L[y, δy] is called the first variation of the functional J [y(x)] on y(x), it is also
called the variation of a functional or variation, it is written as δ J [y(x)], δ J [y] or
δ J , that is
δ J =
x 1
x 0
[F y (x, y, y
)δy + F y (x, y, y
)δy
]dx
=
x 1
x 0
(F y δy + F y δy
)dx =
x 1
x 0
(F y εη + F y εη
)dx
= ε
x 1
x 0
(F y η + F y η
)dx
(2.3.15)
The variational of a functional was a concept posed by Lagrange in 1762. The
variational of a functional is a generalization of the concept of the variation of a
function.
The above definition shows that the variation δ J of the functional J has the
following two properties:
(1) The difference between the increment J and variation δ J of a functional and
is higher order infinitesimal than first order distance d 1 (y, y 1 ), the variation δ J
of a functional is the main linear part of the increment J of a functional (i.e.
main linear part);
(2) The integrand of the variation δ J is a linear function on η and η
.
Using the variation δ J of a functional to express the increment J of a functional
is called the linearization of a functional. The linearization of a functional can be
expressed as J = δ J .
Note that in the variation definition of the above functional, the functions y(x)
and y
(x) both are independent variables, which is different from the differential
definition of a function in the higher mathematics in essence. Table 2.1 shows the
corresponding relations of the functional and function.
Example 2.3.1 Verify J [y] =
x 1
x 0
y
2 dx is not a linear functional.
Solution Since
J [cy] =
x 1
x 0
(cy)
2 dx = c
2
x 1
x 0
y
2 dx = c
x 1
x 0
y
2 dx = c J [y]
so J [y] is not a linear functional.