Math 727: Calculus of Variations
Instructor: Sergey Bolotin
Lectures: MWF, 13:20-14:10, B131 Van Vleck
Office hours: M, 14:30-15:30pm, WF, 15:30-16:30, or by appointment
E-mail:
bolotin@math.wisc.edu
Prerequisites:
Ordinary Differential Equations (Math 319 or better Math 716), Undergraduate Analysis (Math 521-522) or
equivalent.
Math 721 is not a prerequisite, but it will be very helpful.
We'll avoid Lebesgue Integral and Functional Analysis when possible.
Textbooks:
1. G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-dimensional Variational Problems, Oxford Lecture Series
in Mathematics, Vol. 15, OUP (1998).
This will be the main source, but it uses more Analysis than necessary in one-dimensional problems.
We will have less hard Analysis and more Applications.
2. I. M. Gelfand and S. V. Fomin, Calculus of Variations. Dover Publications.
This a classical text unsurpassed in clarity. No prerequisites except rigorous Calculus.
3. L. C. Young. Lectures on the calculus of variations and optimal control theory. Philadelphia, Saunders, 1969.
This is a highly original text by UW professor.
No prerequisites except rigorous Calculus, but not easy reading.
Contains also optimal control theory which we will not touch.
4. V.I.Arnold, Mathematical Methods of Classical Mechanics.
Springer.
This famous book contains applications to classical mechanics.
Course Content:
Calculus of variations is one of the oldest mathematical subjects but still an active field of research.
This course
is mainly about one dimensional calculus of variations where no serious functional analysis is required.
We will start
with necessary conditions for an extremum, Euler-Lagrange equations, theory of second variation, and then proceed to
sufficient conditions and field theory.
In the second part of the course we develop direct methods of
the calculus of
variations and prove existence theorems for minima and other types of critical points.
This requires some basic tools
from functional analysis.
Applications will be chosen from geometry, differential equations and classical mechanics.
Grading:
Will be based on homework. Homework will be
assigned every other Monday and will be due in 2 weeks on Monday.
First HW will be assigned on September 7.
Homework assignments:
Assignment 1.
Assignment 2.
Assignment 3.
Assignment 4.