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
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
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 o
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
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
w
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
(
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
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 th
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 r
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
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 for
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)