MATH 410 Calculus of Variations
Spring 2008
10:50 TTh
Description
In Principia Mathematica Isaac Newton posed the problem of finding
the shape of a body moving along the x-axis which minimized air
resistance. This was the first problem in the calculus of
variations. He solved it in 1694.
Other problems soon followed,
posed and solved by the Bernoulli brothers, Euler and others. Here
are some examples:
- Geodesics. Find the shortest path connecting two points.
If the two points are in the plane you know the answer. If the two
points and the path connecting them are on a sphere you might well
know the answer. In both cases can you prove that your answer is
correct? What if the points and the curve lie on a different surface,
such as the torus?
- Brachistochrone. A particle slides under the force of
gravity along a smooth curve joining the points A and B in the
plane. What is the curve that minimizes the time of transit?
- The path of a ray of light. A light ray proceeds from point
A to point B. Fermat proposed that the path it takes minimizes the
transit time. If the velocity of light is constant in the region
containing A and B, the path is the straight line. What is the path if
the velocity of light is not constant? For example, what if the point
A is in open air, but the point B is under water?
- Heavy chain. Suppose a heavy chain has its ends fixed in
space and is attracted downward by the force of gravity. What is the
shape of the chain?
- Elastic beam. Consider an elastic beam with its ends
clamped at two points. The beam is bent downward by the force of
gravity. What is its shape?
- Minimal surfaces. Given a closed curve in space what is the
shape of the surface with minimal area which has this curve as its
boundary? You can think of the curve as made of wire. The surface of
minimal area is a soap film achieved by dipping the wire in a soap
solution. This is an extremely hard problem. Consider the special
case when the boundary curve consists of two circles centered on the
x-axis lying in planes perpendicular to that axis. Then the surface
is a surface of revolution about the x-axis. What is the curve that
generates that surface?
- Isoperimetric problem. Find the region in the plane with
fixed perimeter L having the largest area. Many of you will realize
that it is a circle, but can you prove it?
All of these problems involve finding extrema where the object being
varied is a function and not a point in Euclidean space, as is the
case in calculus. It is this difference that defines the calculus of
variations. It is clear that the ordinary tools of calculus do not
directly apply. Over time three methods of solution have been
developed:
- Ad hoc methods. Methods depending on the nature of
the specific problems. Sometimes understanding the physics will help
us find the answer and do the math.
- Indirect methods. These methods extend the methods of
the ordinary calculus. As is the case in calculus, we will find
necessary conditions that are not sufficient, and sufficient
conditions that are not necessary.
- Direct methods. Start with a sequence of functions which
come closer and closer to the actual minimum. We have to find out if
the sequence converges or not. If it does converge, does it converges
to a minimum?
In this course we will solve most of the problems listed above, as
well as many others. Although we will spend most of the time solving
problems with one spatial dimension, we will also want to discuss the
situation in several variables. This will allow us to consider
Hamilton's approach to mechanics.
We will occasionally use ad hoc methods, but we will spend
most of our time developing the indirect methods. Hopefully at the
end of the course we will find time to look at some direct methods as
well.
Staff
Instructor
-
John C. Polking
-
Office: HB 450. Office hours: 1:30 to 4:00 on days before homework is
due, Tuesdays and Thursdays immediately after class,
and by appointment.
- Email:
polking@rice.edu
- Telephone: ext 4841 or 713-799-9142
Teaching Assistant
-
Heather Hardway
-
Office: HB 447. Office hours by appointment.
- Email: hardway@rice.edu
- Telephone: ext 2868
Text
The texts for this course are
- Calculus of Variations,
by I.M. Gelfand and S.V. Fomin and
- An Introduction to the Calculus
of Variations by Charles Fox
Both of these are Dover paperbacks
and the total list price is $23.90.
Grading
Half of your final grade for the course will be
determined by your performance on the homework, and the other half on
the final exam.
Homework
There will be a homework assignment each week. The lowest homework grade
will not be counted in determining the grade.
All homework is due in class on the date announced. This will
typically be about a week after the assignment is posted. Each student
will be allowed to have at most one late homework assignment during the
semester. The one late homework will be accepted up to seven days after
the due date, with or without excuse, and without penalty. No other late
homeworks will be accepted even with an excuse. There will be absolutely
no exceptions to these rules.
Many of the exercises extend the material in the
text. Therefore they are as important as the text itself. Many
homework assignments will contain more problems to be done than are to
be turned in. It should be emphasized that a person learns
mathematics by doing problems. You are encouraged to at least look at
all of the exercises in the book.
A homework assignment is meant to convince the grader that you
understand the material. The best way to do that is to use complete
sentences and to organize your work in paragraphs. In your writing,
attempt to achieve the same clarity you find in textbooks. If the
grader cannot understand your writing, your paper will not be
graded.
The homework is not pledged. You are encouraged to discuss the homework,
and to work together on the problems. However each student is responsible
for the final preparation of his or her own homework papers.
Homework Assignments:
John C. Polking <
polking@rice.edu>
Last modified: Tuesday, October 23, 2007