174
2 Variational Problems with Fixed Boundaries
S
s=0
(−1)
i s D
i s F D is u = 0
(2.9.7)
S
s=0
(−1)
i s D
i s
∂ F
∂ D i s u
= 0
(2.9.8)
It should be pointed out that in practical application, the partial derivative of u
with respect to a few independent variables will often appears many times in the
form of different, The total number of terms S should be correctly understood as the
different combinations of the partial derivative of u with respect to the independent
variables.
Proof Taking the first variation to the functional (2.9.4), and taking out the s th
term F D is u δD
i s u of variational of the integrand in it, the term has i s th order partial
derivative, making use of the property that the variational and derivation can be
exchanged the sequence, do integration by parts to it i s times, there is
Ω
F D is u δ D
is udx 1 dx 2 · · · dx m =
Ω
F D is u dδ
∂ i1−1+···+ im u
∂ x
i1−1
1
· · · ∂ x
im
m
dx 2 · · · dx m
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m −
Ω
∂ F D is u
∂ x 1
δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
dx 1 dx 2 · · · dx m
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m −
Ω−x1
∂ F D is u
∂ x 1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m
+ (−1)
2
∂ 2 F D is u
∂ x 2
1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
dx 1 dx 2 · · · dx m = · · ·
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
−
∂ F D is u
∂ x 1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m + · · ·
+ (−1)
2
Ω
∂ 2 F D is u
∂ x 2
1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
dx 1 dx 2 · · · dx m = · · ·
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
−
∂ F D is u
∂ x 1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m + · · ·
+ (−1)
is −1
Ω−xm
∂ is −1 F D is u
∂ x
i1
1 · · · ∂ x
im −1
m
δ
∂u
∂ x m
xm =xm1
xm =xm0
dx 1 dx 2 · · · dx m−1
+ (−1)
is
Ω
∂ is F D is u
∂ x
i1
1 · · · ∂ x
im
m
δudx 1 dx 2 · · · dx m
= B s + (−1)
is
Ω
D
is F D is u δudx 1 dx 2 · · · dx m
where, B s is the sum of all terms associated with boundary integral.
Except the term u, all the other terms of the first variation for the integrand are
done according to the above method, the variations with the partial derivative of u
are all changed into the form of δu, and sum the all terms, which includes the term
F u δu, we obtain
2 Variational Problems with Fixed Boundaries
S
s=0
(−1)
i s D
i s F D is u = 0
(2.9.7)
S
s=0
(−1)
i s D
i s
∂ F
∂ D i s u
= 0
(2.9.8)
It should be pointed out that in practical application, the partial derivative of u
with respect to a few independent variables will often appears many times in the
form of different, The total number of terms S should be correctly understood as the
different combinations of the partial derivative of u with respect to the independent
variables.
Proof Taking the first variation to the functional (2.9.4), and taking out the s th
term F D is u δD
i s u of variational of the integrand in it, the term has i s th order partial
derivative, making use of the property that the variational and derivation can be
exchanged the sequence, do integration by parts to it i s times, there is
Ω
F D is u δ D
is udx 1 dx 2 · · · dx m =
Ω
F D is u dδ
∂ i1−1+···+ im u
∂ x
i1−1
1
· · · ∂ x
im
m
dx 2 · · · dx m
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m −
Ω
∂ F D is u
∂ x 1
δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
dx 1 dx 2 · · · dx m
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m −
Ω−x1
∂ F D is u
∂ x 1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m
+ (−1)
2
∂ 2 F D is u
∂ x 2
1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
dx 1 dx 2 · · · dx m = · · ·
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
−
∂ F D is u
∂ x 1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m + · · ·
+ (−1)
2
Ω
∂ 2 F D is u
∂ x 2
1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
dx 1 dx 2 · · · dx m = · · ·
=
Ω−x1
F D is u δ
∂ i1−1+···+im u
∂ x
i1−1
1
· · · ∂ x
im
m
−
∂ F D is u
∂ x 1
δ
∂ i1−2+···+im u
∂ x
i1−2
1
· · · ∂ x
im
m
x1=x11
x1=x10
dx 2 · · · dx m + · · ·
+ (−1)
is −1
Ω−xm
∂ is −1 F D is u
∂ x
i1
1 · · · ∂ x
im −1
m
δ
∂u
∂ x m
xm =xm1
xm =xm0
dx 1 dx 2 · · · dx m−1
+ (−1)
is
Ω
∂ is F D is u
∂ x
i1
1 · · · ∂ x
im
m
δudx 1 dx 2 · · · dx m
= B s + (−1)
is
Ω
D
is F D is u δudx 1 dx 2 · · · dx m
where, B s is the sum of all terms associated with boundary integral.
Except the term u, all the other terms of the first variation for the integrand are
done according to the above method, the variations with the partial derivative of u
are all changed into the form of δu, and sum the all terms, which includes the term
F u δu, we obtain