326
5 Variational Problems of Conditional Extrema
5.2 Variational Problems with Differential Constraints
The variational problem with differential constraints is discussed in this section, it is
a generalization of the variational problem under the holonomic constraint. Let the
functional
J [y] =
x 1
x 0
F(x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n )dx
(5.2.1)
The constraint conditions are
ϕ i (x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n ) = 0 (i = 1, 2, . . . , m; m < n)
(5.2.2)
The boundary conditions are
y j (x 0 ) = y j0 , y j (x 1 ) = y j1 ( j = 1, 2, . . . , n)
(5.2.3)
The constraint condition (5.2.2) is called the differential constraint, where
ϕ i (i = 1, 2, . . . , m; m < n) are independent of each other. The characteristics
of the differential constraint is that there is the derivative of y in the constraint. At
the moment, the functional (5.2.1) is called the objective functional with differential constraint. Finding the extremal problem of the functional (5.2.1) under the
differential constraints (5.2.2) and the boundary conditions (5.2.3) is also called the
Lagrange problem.
For the extremal problem of the functional under the differential constraints, there
is the following Lagrange theorem similar to Theorem 5.1.1.
Theorem 5.2.1 If under the differential constraints (5.2.2) and the boundary conditions (5.2.3), the objective functional (5.2.1) obtains extremum, then there are the
undetermined functions λ i (x), so that the function y 1 , y 2 , …, y n satisfies the following
auxiliary functional
J
∗
[y] =
x 1
x 0
F +
m
i=1
λ i (x)ϕ i
dx =
x 1
x 0
H dx
(5.2.4)
the corresponding Euler equations are
H y j −
d
dx
H y
j
= 0 ( j = 1, 2, . . . , n)
(5.2.5)
where, H = F +
m
i=1
λ i (x)ϕ i .
Equation (5.2.5) can be written as