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5 Variational Problems of Conditional Extrema
ϕ i (x, y 1 , y 2 , . . . , y n ) = 0 (i = 1, 2, . . . , m; m < n)
(5.1.2)
The boundary conditions are
y j (x 0 ) = y j0 , y j (x 1 ) = y j1 ( j = 1, 2, . . . , n)
(5.1.3)
The solving method of this kind of conditional extremum problem is similar to
the Lagrange multiplier method finding the extremum of the function of several
variables, which can be processed into the unconditional extremum.
The constraint condition (5.1.2) is called the holonomic constraint, geometrical
constraint or finite constraint. The characteristic of the holonomic constraint is
that there are not the derivatives of y in the constraint. At the moment, the functional (5.1.1) is called the objective functional with holonomic constraint. For the
extremal problem of the functional in the holonomic constraint, there is the following
theorem.
Theorem 5.1.1 If under the holonomic constraints (5.1.3) and the boundary conditions (5.1.3), the objective functional (5.1.1) obtains extremum, then there are the
undetermined functions λ i (x), so that the function y 1 , y 2 , …, y n satisfies the following
functional
J
∗
[y] =
x 1
x 0
[F +
m
i=1
λ i (x)ϕ i ]dx =
x 1
x 0
H dx
(5.1.4)
the corresponding Euler equations are
H y j −
d
dx
H y
j
= 0 ( j = 1, 2, . . . , n)
(5.1.5)
where, H = F +
m
i=1
λ i (x)ϕ i . The functional (5.1.4) is called the auxiliary functional.
Theorem 5.1.1 is called the Lagrange theorem. The undetermined function λ i (x)
is called the Lagrange multiplier. Solving the expression of Lagrange multiplier, to
determine its practical significance, this process is called the identifying Lagrange
multiplier or identification of Lagrange multiplier. If the undetermined Lagrange
multiplier in variation is equal to zero, then the variation is called the critical variation. In this case which can’t use the method of undetermined Lagrange multiplier
to incorporate the constraint conditions into the functional, and then to remove the
constraint condition.
When performing the variational operation to the functional (5.1.4), y j , y
j and
λ i (x) should be regarded as the independent functions of the functional J
∗
[y], and
the constraint conditions ϕ i = 0 can be incorporated into the Euler equations of the
functional J
∗
[y] and to consider. Equation (5.1.5) can be written as
5 Variational Problems of Conditional Extrema
ϕ i (x, y 1 , y 2 , . . . , y n ) = 0 (i = 1, 2, . . . , m; m < n)
(5.1.2)
The boundary conditions are
y j (x 0 ) = y j0 , y j (x 1 ) = y j1 ( j = 1, 2, . . . , n)
(5.1.3)
The solving method of this kind of conditional extremum problem is similar to
the Lagrange multiplier method finding the extremum of the function of several
variables, which can be processed into the unconditional extremum.
The constraint condition (5.1.2) is called the holonomic constraint, geometrical
constraint or finite constraint. The characteristic of the holonomic constraint is
that there are not the derivatives of y in the constraint. At the moment, the functional (5.1.1) is called the objective functional with holonomic constraint. For the
extremal problem of the functional in the holonomic constraint, there is the following
theorem.
Theorem 5.1.1 If under the holonomic constraints (5.1.3) and the boundary conditions (5.1.3), the objective functional (5.1.1) obtains extremum, then there are the
undetermined functions λ i (x), so that the function y 1 , y 2 , …, y n satisfies the following
functional
J
∗
[y] =
x 1
x 0
[F +
m
i=1
λ i (x)ϕ i ]dx =
x 1
x 0
H dx
(5.1.4)
the corresponding Euler equations are
H y j −
d
dx
H y
j
= 0 ( j = 1, 2, . . . , n)
(5.1.5)
where, H = F +
m
i=1
λ i (x)ϕ i . The functional (5.1.4) is called the auxiliary functional.
Theorem 5.1.1 is called the Lagrange theorem. The undetermined function λ i (x)
is called the Lagrange multiplier. Solving the expression of Lagrange multiplier, to
determine its practical significance, this process is called the identifying Lagrange
multiplier or identification of Lagrange multiplier. If the undetermined Lagrange
multiplier in variation is equal to zero, then the variation is called the critical variation. In this case which can’t use the method of undetermined Lagrange multiplier
to incorporate the constraint conditions into the functional, and then to remove the
constraint condition.
When performing the variational operation to the functional (5.1.4), y j , y
j and
λ i (x) should be regarded as the independent functions of the functional J
∗
[y], and
the constraint conditions ϕ i = 0 can be incorporated into the Euler equations of the
functional J
∗
[y] and to consider. Equation (5.1.5) can be written as