Appendix: All Solutions to the Problems
801
F =
E A
R
(u − w) 2 +
E I
R 3 (u + w ) 2 − 2R( pu + qw) − p(u 2 + uw ) − q(uw + w 2 )
The various partial derivatives are
F u = −2Rp − 2 pu − pw
− qw
= −2 p(R + u) − w
( p + q),
F u =
2E A
R
(u
− w) +
2E I
R 3 (u
+ w
)
F w = −
2E A
R
(u
− w) − 2Rq, F w = −pu − qu − 2qw
= −( p + q)u − 2qw
,
F w =
2E I
R 3 (u
+ w
)
The Ostrogradsky equations of the functional are
F u =
E A
R
(u
− w
) +
E I
R 3 (u
+ w
) + p(R + u) +
w
( p+q)
2
= 0
E I
R 3 (u
+ w
(4)
) +
( p
+q
)u+( p+q)u
2
+ q
w
+ qw
−
E A
R
(u
− w) − Rq = 0
2.87 Find the Euler equations of the functional J [u, u
∗
]
=
1
2
t 1
t 0
x 1
x 0
[i(u
∗ u t − u
∗
t u) + au
∗ u + bu
∗
x u x + c(u
∗ u)
2
]dxdt.
Solution Let the integrand F = i(u
∗ u t −u
∗
t u)+au
∗ u +bu
∗
x u +c(u
∗ u)
2 , the various
partial derivatives are
F u ∗ = iu t + au + 2cu
∗ u
2
, F u
∗
t
= −iu, F u ∗
x
= bu x
F u = −iu
∗
t + au
∗
+ 2cu
∗2 u, F u t = iu
∗
, F u x = bu
∗
x
The Euler equations of the functional are
2iu t + au + 2cu
∗ u
2
− bu xx = 0
−2iu
∗
t + au
∗
+ 2cu
∗2 u − bu
∗
xx = 0
Chapter 3: Solutions to the Problems in Sufficient
Conditions of Extrema of Functionals
Prove that the extremal curves of the following basic variational problems can be
included in the extremal curve field (proper or central).
3.1 J [y] =
1
0 (y
2
− 2x y)dx, y(0) = y(1) = 0.