Introduction to Applied Math II: Calculus of Variation, Equations of Physics, and Games

5720 Introduction to Applied Math II:
Calculus of Variations, Equations of Physics, and Games

Instructor: Professor Andrej Cherkaev
Office: JWB 225
Telephone: 581-6822
E-mail: cherk@math.utah.edu

Three credit hours.
M,W,F JWB 333
10:45-11:35AM

Addressed to graduate and senior undergraduate students in math, science, and engineering. Grade is based on several homework assignments and a course project.

Text:

Robert Weinstock. Calculus of Variations with Applications to Physics and Engineering. Dover, 1974.

Instructors notes (will be distributed).

Why study extremal problems?

The desire for optimality (perfection) is inherent for humans. The search for extremes inspires mountaineers, scientists, mathematicians, and the rest of the human race. Theory of extremal problems is a mathematical reflection of this noble desire. It formulates and answers the question: What function -- curve or surface -- is the "best?" or: Why there is no "best" solution at all? Deep and elegant methods of Calculus of Variations were originated by Bernoulli, Newton, Euler and actively developed for the last three centuries. The ill-posed problems and problems without solutions are of a special interest: Extremal principles allow for redefinition of a solution itself and restate these problem so that they do possess a unique generalized solution.

Variational problems also root in natural laws of physics. The general principle by Maupertuis proclaims: "If there occur some changes in nature, the amount of action necessary for this change must be as small as possible." This principle is responsible for many philosophical speculations. We use natural variational principles to derive and comment on basic equations of physics and mechanics.

The minimax and Game theories address another basic human passion: Gambling and decision making. This theory, originated in 30-s by Von Neumann and Morgenstern, is now widely used in economics, biology, political and military science.

Preliminary Sillabus:

Basic techniques of Calculus of Variations (2 weeks)

Convexity and symmetrization.
Euler equations, Legendre and Weierstrass tests.
Geometrical optics, brahistohrone, minimal surface of revolution.
Variation of the interval and boundary conditions.

Restricted extremum (2 weeks)

Isoperimentic problem and catenoid problem
Introduction to Optimal control problems.
Duality and Legendre-Young transform.

Hamiltonian mechanics, Symmetries and invariants (1 week)
Multivariable variational problems: (3 weeks)

Reminder of multivariable calculus: Green and Stokes theorems.
Euler-Lagrande equations (minimal surfaces, bubbles),
Differential constraints and integrability, duality.
Stability of solutions and the notion of quasiconvexity
Variation of the domain and of the boundary conditions.

Direct methods and Finite elements.
Variational principles: Do we live in an "optimal" world? (2 weeks)

Conductivity; Stokes flow.
Elasticity.
Eigenvalues and vibrations (Rayleigh principle)
Dissipative media (Minimal principles for complex-valued variables)

Ill-posed variational problems: "Solving" problems that do not have solutions (2 weeks)

Why some problems are ill posed and how to address them?
Regularization and relaxation of ill-posed problems.

Introduction to Game Theory (2 weeks)

Zero-sum and Non-zero-sum games: Strategies and equilibria
Cooperative games and markets

Related URLs:

My Methods of Optimization page
Links and JAVA examples of optimal curves by Alexey Ivanov
Chronology of the game theory by Paul Walker

Class Notes

1. Intoduction, convexity, symmetrization

Webpage about Dido problem.
Webpage about brachistochrone. An interesting prove of its extremal property

2. Euler and Weierstrass tests, Null-Lagrangians.

Home work

The pdf file with the homework is Here