258
4 Problems with Variable Boundaries
2
3
√
9 − x 2 , and ψ
(x 1 ) =
2x 1
3
√
9−x
2
1
. The system of equations about the three unknowns
x 1 , c 1 and c 2 is
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
c 1 + c 2 = 0
c 1 x 1 + c 2 =
1
3
36 − 4x
2
1
1 −
2x 1 c 1
3
9 − x
2
1
= 0
Solving the system of equations, we get
x 1 =
9
5
, c 1 = 2, c 2 = −2
Thus the equation of the extremal curve is y = 2x − 2, and the shortest distance
between the point A(1, 0) and the ellipse 4x
2
+ 9y
2
= 36 is
J [y] =
9
5
1
1 + 2 2 dx =
√
5x
9
5
1
=
4
√
5
5
4.2 Variational Problems of Functionals with Several
Functions
Let the functional of the spatial curve
J [y(x), z(x)] =
x 1
x 0
F(x, y, z, y
, z
)dx
(4.2.1)
where, y, z ∈ C
2
[x 0 , x 1 ], F ∈ C
2 , the admissible lines y = y(x), z = z(x) are
fixed at the left endpoint A(x 0 , y 0 , z 0 ), they are undetermined at the right endpoint
B(x 1 , y 1 , z 1 ).
The variation of the functional (4.2.1) can be done after the method of the above
section. The increment of the functional J [y, z] can be written as
J =
x 1 +δx 1
x 0
F(x, y + δy, z + δz, y
+ δy
, z
+ δz
)dx −
x 1
x 0
F(x, y, z, y
, z
)dx
=
x 1 +δx 1
x 1
F(x, y + δy, z + δz, y
+ δy
, z
+ δz
)dx