UBC Mathematics 402(201)—Assignment 10
Due at 09:00 on Thursday 26 March 2015
1. Consider this isoperimetric problem on a fixed interval, assuming γ > 0:
Z 2
Z 2
p
2
min
x(t) dt = γ .
1 + ẋ(t) dt : x(0) = 0 = x(2),
x
0
0
(a) Prove: If a minimizing arc exists, its graph must be an arc of a circle in the (t, x)-plane.
π
(b) Suppose γ = − 1. Show that a minimizing arc does exist, and describe it in detail.
2
2. Consider the following problem:
Z 1
Z
2
2
min
ẋ(t) − x(t) dt : x(0) = 1, x(1) = e,
0
1
2
2
ẋ(t) + x(t)
0
dt = e − 1 .
2
(a) Find an abnormal extremal (λ0 = 0).
(b) Prove that this extremal provides the problem’s unique global minimum.
(c) Show that there are no admissible extremals for which λ0 = 1.
3. Imagine taking a uniform chain of given length s and fixing its ends at the points (0, 0) and (1, 0).
Gravity will pull the chain into a certain curve: find this curve. In detail,
(a) Present an equation describing the shape of the curve and a clear description of how any
constants in that equation are to be determined; and
(b) Using computer assistance, find and sketch on the same axes the hanging-chain shapes for the
cases s = 1.25, s = 1.50, and s = 2.00.
Suggestions: Assume that s ≥ 1, i.e., there is enough chain to link the given points. Using t as the
label for a horizontal axis in space, and x as the label for a vertical axis, the problem turns into a
search for a particular arc x ∈ P WS[0, 1] satisfying
Z 1p
1 + ẋ(t)2 dt.
(1)
x(0) = 0, x(1) = 0, s =
0
p
Here we recognize ds = 1 + ẋ2 dt as the length of chain lying above the horizontal segment dt.
This short arc of chain will have mass dm = ρ ds (where ρ > 0 is the linear density of the chain,
with units of kg/m), and thus its gravitational potential energy will be
p
dU = (dm) gx = ρgx(t) 1 + ẋ(t)2 dt.
The chain will arrange itself to minimize the total potential,
Z 1
p
def
ρgx(t) 1 + ẋ(t)2 dt.
U [x] =
0
def
Since both ρ and g are positive constants, it suffices to minimize Λ[x] = U [x]/(ρg) subject to the
constraints in line (1).
File “hw10”, version of 19 Mar 2015, page 1.
Typeset at 11:07 March 19, 2015.
4. Use Lagrange Multipliers to find all possible local minimizers in the problem
min
Z
1
2
x(t) dt : x(0) = 0, x(1) = 0,
0
Z
1
ẋ(t) dt = 1 .
2
0
Can convexity methods be used to conclude that a minimum is achieved? Explain.
[Troutman problem 6.23.]
File “hw10”, version of 19 Mar 2015, page 2.
Typeset at 11:07 March 19, 2015.