4.1 Variational Problems of the Simplest Functional
247
Theorem 4.1.3 Let the left endpoint of the extremal curve y = y(x) for the functional J [y(x)] =
x 1
x 0
F(x, y, y
)dx be undetermined on the known curve y = ϕ(x),
while the right endpoint is undetermined on the known curve y = ψ(x), then the
left endpoint at x = x 0 must satisfy
[F + (ϕ
− y
)F y ]
x=x 0
= 0
(4.1.29)
and the right endpoint at x = x 1 must satisfy
[F + (ψ
− y
)F y ]
x=x 1
= 0
(4.1.30)
Finding the most short distance between two known curves on a plane is a common
application of Theorem 4.1.3, in this case, the theorem can be turned into a more
concrete form. The problem is boiled down to finding the minimum of the functional
J [y] =
x 1
x 0
1 + y 2 dx
(4.1.31)
The constraint condition is that the left endpoint of the extremal curve is undetermined on the known curve y = ϕ(x), while the right endpoint is undetermined on
the known curve y = ψ(x). At the moment, Since the integrand is only the function of y
, therefore the general solution the Euler equation of the functional is the
straight line y = c 1 x + c 2 , where c 1 and c 2 are undetermined arbitrary constants,
and y
= c 1 . On the two known curves, the general solution of the Euler equation
has the following form
c 1 x 0 + c 2 = ϕ(x 0 )
(4.1.32)
c 1 x 1 + c 2 = ψ(x 1 )
(4.1.33)
where, x 0 is the intersection point of the extremal curve of the functional and the
function y = ϕ(x); x 1 is the intersection of the extremal curve of the functional ane
the function y = ψ(x).
For the functional (4.1.31), the transversality conditions have the following form
1 + y 2 + (ϕ
− y
)
y
1 + y 2
x=x 0
= 0
(4.1.34)
1 + y 2 + (ψ
− y
)
y
1 + y 2
x=x 1
= 0
(4.1.35)