Amath 507
Homework Solution #1.6.1
Problem: Descent time down a cycloidal curve).
Show that the descent time down the cycloidal curve
x( ) = a + R ( sin ) , y( ) = y a R (1 cos )
(1.1)
is
T =
R
,
g b
(1.2)
where b is the angle corresponding to the point
Amath 507
Homework Solution #3.6.5
(Extra Credit)
The Terrestrial Brachistochrone
If we choose as our independent
variable, our integral,
T =
(r) 2 + r 2
d ,
R2 r 2
R
g
1 1
2 R u
d =
Substitution 2:
Now, Ill substitute for the ugliest term
in town. Il
Amath 507
Homework Solution #1.6.4
Problem: (Potential energy inside the earth).
Use your results from Exercise 1.6.3 and integrate over shells of
appropriate radii to show that the potential energy of a point mass m in a
spherical and homogeneous earth c
Amath 507
Homework Solution #1.6.4
Problem: Potential energy inside the earth
Use your results from Exercise 1.6.3 and integrate over shells of appropriate
radii to show that the potential energy of a point mass m in a spherical and homogeneous earth can
Amath 507
Homework Solution #1.6.3
Problem: Potential energy due to a spherical shell
The gravitational potential energy between two point masses, M and m, separated by a distance r is
GMm
,
(3.1)
r
where G is the universal gravitational constant.
Calcula
Amath 507
Homework Solution #1.6.1
Problem: Descent time down a cycloidal curve
Show that the descent time down the cycloidal curve
x( ) = a + R ( sin ) , y( ) = y a R (1 cos )
(1.1)
is
T =
R
,
g b
(1.2)
where b is the angle corresponding to the point B
Amath 507
Homework Solution #9.6.6
Problem: A Transversality Condition
Find the extremals for the functional
b
J[y] =
0
1 + y2
y
dx
subject to the boundary condition
y(0) = 0
and to the condition that the right endpoint,
(b, y b ), can move along the circ
AMATH 507
HOMEWORK SOLUTION #8.6.2
Question:
Formulate the problem of the minimum surface of revolution as a homogeneous
problem. Why might one of the two possible EulerLagrange equations be better than
the other ? Find the extremal arc for this problem.
Amath 507
Homework Solution #3.6.4
Problem (Brachistochorne on a cylinder):
Find the minimum-time curve of descent of a particle, under the inuence of gravity, on a vertical circular cylinder of xed radius r. Assume that the particle starts from
rest, tha
Amath 507
Homework Solution #3.6.1
Problem:
Find the extremals for the following functionals. Most of the extremals are wellknown geometric curves. For each extremal, name, draw, or describe the curve.
Answers:
1.
The functional
b
a
F[y(x)] =
1 + (y)2
y
d
Amath 507
Homework Solution #2.5.1
Problem (Eulers approach):
Using Eulers approach from Section 2.2, determine polygonal
approximations to the curve that minimizes
2
[(y)
2
+ 6x 2 y] dx
(1.1)
0
subject to
y(0) = 2, y(2) = 4
(1.2)
for n = 1, n = 2, and n
Amath 507
Homework Solution #1.6.10
Problem: (Scherks minimal surface).
Take the minimal surface equation,
2
2
(1 + f y ) f xx 2 f x f y f xy + (1 + f x ) f yy = 0 ,
and look for a solution of the form
f (x, y) = g(x) + h(y) .
(10.1)
Show that the resulti
Amath 507
Homework Solution #1.6.3
Problem: (Potential energy due to a spherical shell).
The gravitational potential energy between two point masses, M and m, separated by a distance r is
GMm
,
(3.1)
r
where G is the universal gravitational constant.
Calc