MA591x , FALL 2004
Introduction to the
Calculus of Variations
Days: M W F. Time: Arranged. Bldg: Harrelson, Rm: Arranged
Instructor: X. Lin
Text book: Introduction to the Calculus of Variations and its
Applications
by Frederick Y.M. Wan
The grade will be determined by one midterm test and one project.
Calculus of variations deal with problems in physics, engineering and
applied mathematics that are governed by maximum or minimum principles.
It can be viewed as a generalization of finding extremal problems in
calculus. However, the minimizer is not a number but an unknown
function. The differential equation for the system can be derived as
critical point for some quantity, e.g., energy. The differential
equation for the unknown function is called the Euler-Lagrange equation.
Examples are abundant in physics, electromagnetic and mechanical
systems. Recently, new applications on biological systems have been
discovered. A classical problem is the so called ``Brachistochrone'':
Find a curve connecting two spatial points along which a particle can
slide down in the shortest time. The other is ``Plateau's problem'':
Find the generator of a revolutional surface with minimal surface
area. In biology, plants must find a way to use incoming
resources to maximize the chance of survive. It can be shown that the
resource must be used to grow the body mass first then for the
reproduction after certain age.
Calculus of variations is closely related to many areas of applied
mathematics, e.g., control theory, Hamilton-Jacobi's theory, and the
finite element method. This course can prepare the students for related
more advance courses.
This is an introductory course accessible to advance undergraduate and
beginning graduate students. Students should have a good back ground in
ordinary differential equations and multivariable calculus. They must
have one semester advanced calculus to understand ``epsilon-delta'' and
``uniform convergence''.
