Problem set 6 due Wednesday June 27 by 2:45 pm in class or in the mailbox.
Problem 1.
Find the function x(t) that maximizes the integral
2
Z
−
1
ẋ2
dt
t2
with boundary conditions given by x(1) = 0 and x(2) = 3.
Problem 2.
Find the function x(t) that minimizes the integral
Z π
ẋ2 + x2 + ẋ cos(2t) dt
0
with boundary conditions given by x(0) = 0 and x(π) = 1.
Problem 3.
We want to find the function x(t) that minimizes the integral
1
Z
ẋ2 dt
0
subject to the integral constraint
Z
1
x dt = 1
0
and with boundary conditions given by x(0) = 0 and x(1) = 0.
(a) Solve (in x(t), λ) the Euler equation corresponding to this problem.
(b*) I did not give in class for problems with integral constraint any convexity statement to show that a function solving the Euler equation needs to be an optimal function. Can you guess (by analogy with optimization
problems for functions of a finite number of variables with equality constraints) a correct statement and show it
applies to the case we’re considering ?
(c) Comment on the sign of the Lagrange multiplier (use the interpretation of Lagrange multiplier we saw for
functions of a finite number of variables).
Problem 4. Soap bubbles
Films of soaps take shapes that minimize their surface area. We’re trying to understand what shape a soap
bubble attached to two metal circles will take.
We have to metal circles C0 = {(0, y, z), y 2 + z 2 = 1} and Cb = {(b, y, z), y 2 + z 2 = B 2 }. The two circles are
hence in parallel planes, centered around the same axis y = z = 0, at distance b of each other, and of respective
radii 1 and B.
A soap bubble based to these circles will take a shape that is rotationally invariant around the axis y = z = 0.
Hence the whole shape of such a bubble can be described by a single function r(x) such that r(0) = 1 and
r(b) = B. The soap film is then the collection of circles Cx = {(x, y, z), y 2 + z 2 = r(x)2 } for 0 ≤ x ≤ b.
Recall that the infinitesimal length of a curve r(x) is given via Pythagoras’ theorem by
p
p
dl = dx2 + dr2 = 1 + r0 (x)2 dx
1
Hence the area of a soap bubble is given by
Z
A(r) = 2π
b
p
r(x) 1 + r0 (x)2 dx
0
The optimization problem we thus have is to minimize A(r) with boundary conditions r(0) = 1 and r(b) = B.
(a) Show that the functions
r(x) = c cosh
x
+d
c
solve the Euler equation (where c and d are two real parameters).
p
Hint : F (x, r, r0 ) = r(x) 1 + r0 (x)2 doesn’t have any explicit dependency on x, so you can use a special
version of the Euler equation. Remember that cosh(x)2 = sinh(x)2 + 1.
(b) Using one of the boundary condition, show that
c=
1
cosh(d)
(c) We now want to try and see if we can find a solution, depending on the value of the second boundary
condition.
Show that for all real number x, cosh(x) ≥ |x|.
Hint : Only consider x ≥ 0. Find the minimum of f (x) = cosh(x) − x. Keep in mind the fact that
cosh(x)2 = sinh(x)2 + 1.
(d) Use (c) to show that
B≥
b cosh(d) + d
cosh(d)
Use (c) again to show that B ≥ b − 1.
(e) Suppose B < b − 1. What is the mathematical conclusion ? What does this mean physically ?
Hint : Fix the radius B. Hold the two circles in your hands and let the distance b between them grow. What
is going to happen to the soap film ?
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