APM462: Homework 5
Due date: Wed Apr 8, 9pm.
(1) Let A = {u : [a, b] 7→ R | u ∈ C
1
, u(a) = A, u(b) = B, u ≥ 0}.
Let F[u(·)] be the volume of the surface of revolution of u formed
by rotating u(x) around the x-axis. Find the u∗ whose surface of
revolution has minimal volume among all u ∈ A. It’s enough to find
the Euler-Lagrange equation without solving it. Hint: compare with
example on p.4 of the notes.
(2) Let A := {u = (u1, u2, u3) : [0, 1] → R
3
| u ∈ C
1} and consider the
“holonomic constraints” problem:
minimize F[u(·)] := Z 1
0
q
u
0
1
(t)
2 + u
0
2
(t)
2 + u
0
3
(t)
2 dt
subject to: u ∈ A, G(u1(t), u2(t), u3(t)) := u1(t)
2 + u2(t)
2 − 1 ≡ 0.
Find, but do not solve, the Euler-Lagrange equations for this problem.
(3) Let A := {u = (u1, u2) : [0, 1] → R
2
| u ∈ C
1} and consider the
“holonomic constraints” problem:
minimize F[u(·)] := Z 1
0
q
u
0
1
(t)
2 + u
0
2
(t)
2 dt
subject to: u ∈ A, G(u1(t), u2(t)) := u1(t) + u2(t)
2 − 1 ≡ 0.
(a) Thinking of this problem as a holonomic problem, find the
Euler-Lagrange equations (you do not need to solve them).
(b) Formulate the above problem as a problem without constraints
and solve it. Hint: note that the holonomic constraint is just
saying that u1(t) = 1 − u2(t)
2
.
(c) Show that parts (a) and (b) give the same answer. Hint: what
is λ(t)?
(4) Recall the optimization problem on Q.5, HW4:
minimize I[x(·)] = 1
2
Z π
0
x
0
(t)
2
dt
subject to the conditions x(0) = x(π) = 0 and the constraint
J[x(·)] = Z π
0
x(t)
2
dt = 1.
(a) Treating this problem as an “isoperimetric problem”, find the
1
st order neccessary conditions for a minimizer.
1
2
(b) What is the relationship between the Lagrange multiplier λ you
found in part (a) above and the λ you found in Q.5(c) on HW4?
(5) Consider the following problem where α is a constant:
minimize F[u(·)] := Z 1
0
L
F
(x, u(x), u0
(x)) dx
subject to: G[u(·)] = Z 1
0
u
0
(x) dx = α;
where u(·) ∈ A := {C
1
functions u on [0, 1] such that u(0) = 0}.
Note that the space A has only one end point condition on u.
(a) Formulate the above problem as a problem without constraints
and solve it. Hint: by integrating the constraint G[u(·)] = α we
see it is equivalent to the condition u(1) = α.
(b) Treating this problem as an isoperimetric problem, find the
Euler-Lagrange equation.
(c) Show that parts (a) and (b) give the same answer.
(6) Consider the following problem (a variant of Q.6) where α is a constant and g : [0, 1] 7→ [0, 1] is a C
1
invertible function:
minimize F[u(·)] := Z 1
0
L
F
(x, u(x), u0
(x)) dx
subject to: G[u(·)] = Z 1
0
g
0
(u(x))u
0
(x) dx = α;
where u(·) ∈ A := {C
1
functions u on [0, 1]}.
Note that here, unlike Q.6, the space A has no conditions on u at
the end points. Find, but do not solve, the first oreder conditions
for this problem.
Hint: formulate the problem as a problem without constraints but
with a condition at one end point. Now use a variant of Q.3 HW4
where there is one end point condition on u. Your answer should
include an Euler-Lagrange equation and one other condition.

EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

E-mail: easydue@outlook.com  微信:easydue

EasyDue™是一个服务全球中国留学生的专业代写公司