4.2 Variational Problems of Functionals with Several Functions
261
Similarly, let Φ y 1 = 0, then δy 1 = −(Φ x 1 /Φ y 1 )δx 1 − (Φ z 1 /Φ y 1 )δz 1 can be solved,
substituting it into Eq. (4.2.7), we get
F − y
F y − z
F z − F y
Φ x
Φ y
x=x 1
δx 1 +
F z − F y
Φ z
Φ y
x=x 1
δz 1 = 0
Since δx 1 and δz 1 are arbitrary, therefore the natural boundary condition is
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
F − y
F y − z
F z − F y
Φ x
Φ y
x=x 1
= 0
F z − F y
Φ z
Φ y
x=x 1
= 0
(4.2.12)
Let Φ z 1 = 0, then δz 1 = −(Φ x 1 /Φ z 1 )δx 1 − (Φ y 1 /Φ z 1 )δy 1 can be solved,
substituting it into Eq. (4.2.7), we get
F − y
F y − z
F z − F z
Φ x
Φ z
x=x 1
δx 1 +
F y − F z
Φ y
Φ z
x=x 1
δy 1 = 0
Since δx 1 and δy 1 are arbitrary, therefore the natural boundary condition is
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
F − y
F y − z
F z − F z
Φ x
Φ z
x=x 1
= 0
F y − F z
Φ y
Φ z
x=x 1
= 0
(4.2.13)
(4) If boundary point B(x 1 , y 1 , z 1 ) can change on the spatial plane x = x 1 , then
δx 1 = 0, δy 1 and δz 1 can be arbitrary values, the natural boundary condition is
F y
x=x 1
= 0, F z | x=x 1 = 0
(4.2.14)
Theorem 4.2.1 Let the boundary conditions y(x 0 ) = y 0 , z(x 0 ) = z 0 at
the left endpoint of the extremal curve for the functional J [y(x), z(x)] =
x 1
x 0
F(x, y, z, y
, z
)dx be fixed, while another endpoint B(x 1 , y 1 , z 1 ) changes on
the known curves y 1 = ϕ(x 1 ), z 1 = ψ(x 1 ), then the extremal curve y = y(x) of the
variable endpoint must satisfy the transversality condition (4.2.9).
Corollary 4.2.1 If boundary point A(x 0 , y 0 , z 0 ) changes on the known curves
y 0 = ϕ(x 0 ), z 0 = ψ(x 0 ), then the extremal curve of the functional J [y, z] =
x 1
x 0
F(x, y, z, y
, z
)dx must satisfy the transversality condition at boundary point
A(x 0 , y 0 , z 0 )
[F + (ϕ
− y
)F y + (ψ
− z
)F z ]
x=x 0
= 0
(4.2.15)