4.2 Variational Problems of Functionals with Several Functions
269
J [y, z] =
1
x 1
1 + y 2 + z 2 dx
(1)
where, point B(x 1 , y 1 , z 1 ) at which x 1 is should be on the sphere x
2
+ y
2
+ z
2
= 1.
The extremal curve the functional (1) is the straight line
y = c 1 x + c 2
z = c 3 x + c 4
(2)
The extremal curve (2) should pass through point A(1, 1, 1), there is
c 1 + c 2 = 1
c 3 + c 4 = 1
(3)
The transversality conditions at B(x 1 , y 1 , z 1 ) are
1 + y 2 + z 2 −
y 2
1 + y 2 + z 2
+
−x
1 − x 2 − y 2
− z
z
1 + y 2 + z 2
x=x 1
= 0
(4)
y
1 + y 2 + z 2
−
y
1 − x 2 − y 2
z
1 + y 2 + z 2
x=x 1
= 0
( 5 )
Simplifying Eqs. (4) and (5), and taking note that y
= c 1 , z
= c 3 , there are
z 1 − c 3 x 1 = 0
c 1 z 1 − c 3 y 1 = 0
(6)
According to the condition that the extremal curve should pass through point
B(x 1 , y 1 , z 1 ), there are
y 1 = c 1 x 1 + c 2
z 1 = c 3 x 1 + c 4
(7)
Making use of the spherical equation, solving simultaneously Eqs. (3), (6) and
(7), we get c 1 = c 3 = 1, c 2 = c 4 = 0, thus the equation of the extremal curve is
y = x
z = x
(8)
Because point B(x 1 , y 1 , z 1 ) is on the sphere, using the above equations we get
x 1 = ±
√
3
3
, therefore the two points can be obtained