Appendix: All Solutions to the Problems
849
where, the integral constants c 1 , c 2 and c 3 are determined by the boundary
conditions.
5.10 Find the shortest distance between two points A(1, −1, 0) and B(2, 1, −1) on
the surface 15x − 7y + z − 22 = 0.
Solution The distance between two points on the surface is the following functional
J =
2
1
1 + y 2 + z 2 dx
(1)
The constraint condition is
ϕ(x, y, z) = 15x − 7y + z − 22 = 0
( 2 )
Establishing the auxiliary functional
J
∗
=
x 1
x 0
[
1 + y 2 + z 2 + λ(x)(15x − 7y + z − 22)]dx
(3)
The Euler equations of the functional are
−7λ(x) −
d
dx
y
1 + y 2 + z 2
= 0
( 4 )
λ(x) −
d
dx
z
1 + y 2 + z 2
= 0
( 5 )
Eq. (5) is multiplied by 7, adding Eq. (5) and Eq. (4) and integrating, we obtain
y
+ 7z
1 + y 2 + z 2
= c
(6)
From the constraint condition (2), we get
z
= 7y
− 15
(7)
Substituting Eq. (7) into Eq. (6), we get
y
+ 7(7y
− 15)
1 + y 2 + (7y − 15) 2
= c
(8)
namely y
= c 1 , integrating we get y = c 1 x + c 2 , from the endpoint conditions,
we get c 1 = 2, c 2 = −3, hence