2.3 Variations of the Simplest Functionals and Necessary Conditions …
103
In this way the increment of the simplest functional J [y] =
x 1
x 0
F(x, y, y
)dx
can also be expressed as
J =
x 1
x 0
(F y δy + F y δy
)dx +
x 1
x 0
R 1 dx =
x 1
x 0
(F y δy + F y δy
)dx + εd 1 [y 1 , y]
(2.3.13)
where
εd 1 [y 1 , y] =
x 1
x 0
R 1 dx
(2.3.14)
and ε approaches zero with d 1 [y 1 , y] approaching zero.
It is observed from Eqs. (2.3.8) and (2.3.13) that the difference between
x 1
x 0
(F y δy + F y δy
)dx and J is a higher order infinitesimal than d 1 [y 1 , y],
x 1
x 0
(F y δy + F y δy
)dx is the principal part of the increment of a functional, it is
written as L[y, δy]. Later that L[y, δy] is a linear functional about δy will be further
proved. Below the definition of the linear functional is given.
If a continuous functional J [y(x)] satisfies the following two conditions:
(1) J [y 1 (x) + y 2 (x)] = J [y 1 (x)] + J [y 2 (x)]
(2) J [cy(x)] = c J [y(x)]
where, c is an arbitrary constant, then J [y(x)] is called the linear functional on
y(x).
Since
L[y, δy 1 + δy 2 ] =
x 1
x 0
[F y (δy 1 + δy 2 ) + F y (δy
1 + δy
2 )]dx
=
x 1
x 0
[F y δy 1 + F y δy
1 ]dx +
x 1
x 0
[F y δy 2 + F y δy
2 ]dx
= L[y, δy 1 ] + L[y, δy 2 ]
L[y, cδy] =
x 1
x 0
[F y (cδy) + F y (cδy
)]dx = c
x 1
x 0
(F y δy + F y δy
)dx = cL[y, δy]
so L[y, δy] is the linear functional on δy.
If a continuous functional J [u, v] satisfies the following three conditions:
(1) J [u, v] = J [v, u]
(2) J [a 1 u 1 + a 2 u 2 , v] = a 1 J [u 1 , v] + a 2 J [u 2 , v]
(3) J [u, b 1 v 1 + b 2 v 2 ] = b 1 J [u, v 1 ] + b 2 J [u, v 2 ]