Homework Solution #9.6.6
Problem: A Transversality Condition
Find the extremals for the functional
1 + y2
subject to the boundary condition
y(0) = 0
and to the condition that the right endpoint,
(b, y b ), can move along the circ
HOMEWORK SOLUTION #8.6.2
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.
Homework Solution #3.6.5
The Terrestrial Brachistochrone
If we choose as our independent
variable, our integral,
(r) 2 + r 2
R2 r 2
2 R u
Now, Ill substitute for the ugliest term
in town. Il
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
Homework Solution #3.6.1
Find the extremals for the following functionals. Most of the extremals are wellknown geometric curves. For each extremal, name, draw, or describe the curve.
1 + (y)2
Homework Solution #2.5.1
Problem (Eulers approach):
Using Eulers approach from Section 2.2, determine polygonal
approximations to the curve that minimizes
+ 6x 2 y] dx
y(0) = 2, y(2) = 4
for n = 1, n = 2, and n
Homework Solution #1.6.10
Problem: (Scherks minimal surface).
Take the minimal surface equation,
(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) .
Show that the resulti
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
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
where G is the universal gravitational constant.
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 )
where b is the angle corresponding to the point