270
4 Problems with Variable Boundaries
B 1
√
3
3
,
√
3
3
,
√
3
3
and B 2
−
√
3
3
, −
√
3
3
, −
√
3
3
On the extremal curve (8) jointing point A and point B 1 , the functional (1) obtains
the minimum, the minimum is
J min =
1
√
3
3
1 + 1 2 + 1 2 dx =
√
3 − 1
( 9 )
On the extremal curve (8) jointing point A and point B 2 , the functional (1) obtains
the maximum, the maximum is
J max =
1
−
√
3
3
1 + 1 2 + 1 2 dx =
√
3 + 1
( 1 0 )
4.3 Variational Problems of Functionals with Higher Order
Derivatives
This section discusses the extremal problem with the functional of higher derivative,
the boundary points of the functional can change. Here first to discuss the higherorder derivative is the situation of the second order, then to discuss the functional
with the higher derivative.
4.3.1 Cases of Functionals with One Unknown Function
and Its Second Derivative
Let the functional
J [y(x)] =
x 1
x 0
F(x, y, y
, y
)dx
(4.3.1)
where, y ∈ C
4
[x 0 , x 1 ], F ∈ C
3 , the admissible function y = y(x) is fixed at the left
endpoint A(x 0 , y 0 ), and it is variable at the right endpoint B(x 1 , y 1 ). The extremal
curve of the functional (4.3.1) must satisfy the Euler-Poisson equation