262
4 Problems with Variable Boundaries
Corollary 4.2.2 If boundary point B(x 1 , y 1 , z 1 ) changes along a surface z 1 =
ϕ(x 1 , y 1 ), then the extremal curves y = y(x), z = z(x) of the functional (4.2.1)
must satisfy the transversality conditions at B(x 1 , y 1 , z 1 )
[F − y
F y + (ϕ x − z
)F z ]
x=x 1
= 0
(F y + F z ϕ y )
x=x 1
= 0
(4.2.16)
The two conditions with z 1 = ϕ(x 1 , y 1 ) can determine the four arbitrary constants
in general solution of the Euler equations.
Corollary
4.2.3 Let
the
functional
J [y 1 , y 2 , . . . , y n ]
=
x 1
x 0
F(x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n )dx, then the boundary conditions can
be obtained in the case that endpoint B(x 1 , y 11 , y 21 , . . . , y n1 ) is variable
⎧
⎪ ⎨
⎪ ⎩
F −
n
i=1
y
i F y
i
x=x 1
= 0
F y
i
x=x 1
= 0
(i = 1, 2, . . . , n)
(4.2.17)
Corollary
4.2.4 Let
the
functional
J [y 1 , y 2 , . . . , y n ]
=
x 1
x 0
F(x, y 1 , y 2 , . . . , y n , y
1 , y
2 , . . . , y
n )dx,
if
boundary
point
B(x 1 , y 11 , y 21 , . . . , y n1 ) changes on the known curve y i = ψ i (x 1 ) (i = 1, 2, . . . , n),
then the extremal function at boundary point B(x 1 , y 11 , y 21 , . . . , y n1 ) must satisfy
the transversality condition
F +
n
i=1
(ψ
i − y
i )F y
i
x=x 1
= 0
(4.2.18)
If endpoint A is also variable, then it is same treated as endpoint B.
Example 4.2.1 Find the transversality condition of the functional J [y, z] =
x 1
x 0
f (x, y, z)
1 + y 2 + z 2 dx, point B(x 1 , y 1 , z 1 ) changes on the surface z 1 =
ϕ(x 1 , y 1 ).
Solution According to Corollary 4.2.2, when x = x 1 , the transversality conditions
are
[F − y
F y + (ϕ x − z
)F z ]
x=x 1
= 0
(F y + F z ϕ y )
x=x 1
= 0
consequently
1 + ϕ x z
= 0, y
+ z
ϕ y = 0
When x = x 1 , the above two equations are merged into