3.6 Higher Order Variations of Functionals
229
where, ¯
F yy , ¯
F yy and ¯
F y y denote the values of F yy , F yy and F y y at point (x, y +θ 1 δy,
y
+ θ 2 δy
), 0 < θ 1 < 1, 0 < θ 2 < 1.
According to the continuity of ¯
F yy , ¯
F yy and ¯
F y y , when d 1 [y, y+δy] is sufficiently
small, there are
¯
F yy = F yy + ε 1 , ¯
F yy = F yy + ε 2 , ¯
F y y = F y y + ε 3
(3.6.3)
where, ε 1 , ε 2 and ε 3 all approaches zero with d 1 [y, y + δy] → 0. Thus F can be
written as the following form
F = F(x, y + δy, y
+ δy
) − F(x, y, y
)
= (F y δy + F y δy
) +
1
2
[F yy (δy)
2
+ 2F yy δyδy
+ F y y (δy
)
2
] + ε (3.6.4)
where
ε =
1
2
[ε 1 (δy)
2
+ 2ε 2 δyδy
+ ε 3 (δy
)
2
]
(3.6.5)
Below to prove that ε is the higher order infinitesimal than d
2
1 [y, y + δy].
Since (δy − δy
)
2
≥ 0, namely
2δyδy
≤ (δy)
2
+ (δy
)
2 , so there is
ε =
1
2
[ε 1 (δy)
2
+ 2ε 2 δyδy
+ ε 3 (δy
)
2
] ≤
1
2
[(|ε 1 | + |ε 2 |)(δy)
2
+ (|ε 2 | + |ε 3 |)(δy
)
2
]
=
1
2
[ε 4 (δy)
2
+ ε 5 (δy
)
2
]
(3.6.6)
where, ε 4 = |ε 1 |+|ε 2 | and ε 5 = |ε 2 |+|ε 3 | both approaches zero with d 1 [y, y+δy] →
0. Moreover since |δy| ≤ d 1 [y, y + δy],
δy
≤ d 1 [y, y + δy], so that
|ε| ≤
1
2
[ε 3 (δy)
2
+ ε 4 (δy
)
2
] ≤
1
2
[ε 3 + ε 4 ]d
2
1 [y, y + δy]
(3.6.7)
This shows that ε is the higher order infinitesimal than d
2
1 [y, y + δy].
The first term on the right side for the expression (3.6.4) is called the first variation
of the function F(x, y, y
), it is written as δ F, that is
δ F = F y δy + F y δy
(3.6.8)
The second term on the right side for the expression (3.6.4) is called the quadratic
variation or second variation of the function F(x, y, y
), it is written as δ
2 F, that
is
δ
2 F =
1
2
[F yy (δy)
2
+ 2F yy δyδy
+ F y y (δy
)
2
]
(3.6.9)