4.1 Variational Problems of the Simplest Functional
245
condition. The essential boundary condition is also called the Dirichlet boundary
condition, and the natural boundary condition is also called the Neumann boundary
condition. The natural boundary condition is the variational condition of a functional
on the boundary of domain. It can be seen that in calculus of variations there are two
kinds of different boundary conditions in property. Because the natural boundary
condition is not given in advance, but is automatically satisfied by the extremal
function y(x), and so is not listed as a definite condition.
Theorem 4.1.1 Let the one end of the extremal curve y = y(x) of the functional
J [y(x)] =
x 1
x 0
F(x, y, y
)dx be fixed, while the other end is undetermined on straight
line x = x 1 , then the undetermined end must satisfy the natural boundary condition
(4.1.24).
If the endpoint of the extremal curve y = y(x) is undetermined on the known
curve y = ψ(x), then the variation δx 1 is related to δy 1 .
Example 4.1.1 Find the natural boundary condition of extremal problem for the
functional J [y] =
x 1
x 0
[ p(x)y
2
+ q(x)y
2
+ 2 f (x)y]dx, where x 0 and x 1 are both
the free boundary, p(x), q(x) and f (x) are all known functions, and p(x) = 0.
Solution Since x 0 and x 1 are both the free boundary, according to Theorem 4.1.1,
the natural boundary conditions are
F y
x=x 0
= 2 p(x)y
x=x 0
= 0, F y
x=x 1
= 2 p(x)y
x=x 1
= 0
Because of p(x) = 0, the natural boundary conditions can be changed into
y
x=x 0
= y
(x 0 ) = 0, y
x=x 1
= y
(x 1 ) = 0
Theorem 4.1.2 Let the left endpoint of the extremal curve y = y(x) for the functional J [y(x)] =
x 1
x 0
F(x, y, y
)dx be fixed, while the right endpoint is undetermined
on the known curve y = ψ(x), then the right endpoint must satisfy at x = x 1
[F + (ψ
− y
)F y ]
x=x 1
= 0
(4.1.25)
Proof Taking the variation to the known curve y = ψ(x), we get
δy = ψ
(x)δx
Because x and y are at the right endpoint, x = x 1 , y = y 1 , from Eq. (4.1.22), we
obtain
[F + (ψ
− y
)F y ]
x=x 1
δx 1 = 0
245
condition. The essential boundary condition is also called the Dirichlet boundary
condition, and the natural boundary condition is also called the Neumann boundary
condition. The natural boundary condition is the variational condition of a functional
on the boundary of domain. It can be seen that in calculus of variations there are two
kinds of different boundary conditions in property. Because the natural boundary
condition is not given in advance, but is automatically satisfied by the extremal
function y(x), and so is not listed as a definite condition.
Theorem 4.1.1 Let the one end of the extremal curve y = y(x) of the functional
J [y(x)] =
x 1
x 0
F(x, y, y
)dx be fixed, while the other end is undetermined on straight
line x = x 1 , then the undetermined end must satisfy the natural boundary condition
(4.1.24).
If the endpoint of the extremal curve y = y(x) is undetermined on the known
curve y = ψ(x), then the variation δx 1 is related to δy 1 .
Example 4.1.1 Find the natural boundary condition of extremal problem for the
functional J [y] =
x 1
x 0
[ p(x)y
2
+ q(x)y
2
+ 2 f (x)y]dx, where x 0 and x 1 are both
the free boundary, p(x), q(x) and f (x) are all known functions, and p(x) = 0.
Solution Since x 0 and x 1 are both the free boundary, according to Theorem 4.1.1,
the natural boundary conditions are
F y
x=x 0
= 2 p(x)y
x=x 0
= 0, F y
x=x 1
= 2 p(x)y
x=x 1
= 0
Because of p(x) = 0, the natural boundary conditions can be changed into
y
x=x 0
= y
(x 0 ) = 0, y
x=x 1
= y
(x 1 ) = 0
Theorem 4.1.2 Let the left endpoint of the extremal curve y = y(x) for the functional J [y(x)] =
x 1
x 0
F(x, y, y
)dx be fixed, while the right endpoint is undetermined
on the known curve y = ψ(x), then the right endpoint must satisfy at x = x 1
[F + (ψ
− y
)F y ]
x=x 1
= 0
(4.1.25)
Proof Taking the variation to the known curve y = ψ(x), we get
δy = ψ
(x)δx
Because x and y are at the right endpoint, x = x 1 , y = y 1 , from Eq. (4.1.22), we
obtain
[F + (ψ
− y
)F y ]
x=x 1
δx 1 = 0