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

Course's web page: www.math.utah.edu/~cherk/teach/appl-math2002.html

Text:

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

Internet notes: (URL will be posted)

Instructors notes (will be distributed).

Recommended advanced reading:

1. I.M. Gelfand and S.V. Fomin. Calculus of variations. Prentice-Hall, 1963.
2. Andrej Cherkaev. Variational Methods for Structural Optimization. Springer NY, 2000.
3. R. Duncan Luce and Howard Raiffa. Games and Decisions. Dover, 1989.
4. Herbert Gintis. Game Theory Evolving. Princeton University Press,  2000.

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.
 
 

• Basic techniques of Calculus of Variations (2 weeks)
• Some classical inequalities
• Euler equations (classical examples: brahistohrone and geometrical optics, minimal surface of revolution)
• Restricted problems (isoperimentic problem and catenoid problem)
• Legendre and Weierstrass tests (strong variation, distinguishing max from min)
• Variation of the interval and boundary conditions
• Hamiltonian: Symmetries and invariants (conservation laws) (1 week)
• Duality and Legendre-Young transform. (1 week)
• Multivariable variational problems: (2 weeks)
• Reminder of multivariable calculus: Green and Stokes theorems.
• Euler-Lagrande equations (minimal surfaces, bubbles),
• Differential constraints and integrability, duality.
• Direct methods and Finite elements.
• Variational principles: (3 weeks) (Do we live in an "optimal"  world?)
• Conductivity; Stokes flow.
• Elasticity.
• Eigenvalues and vibrations (Rayleigh principle)
• Dissipative media (Minimal principles for complex-valued variables)
• Schrodinger equations
• Ill posed variational problems: "Solving"  problems without solutions (2 weeks)
• Why some problems are ill posed and how to address them?
• Applications: Structural optimization and phase transitions.
• Inverse problems and Regularization
•  Introduction to Games and minimax problems (2 weeks)
•  Zero-sum and Non zero-sum games (poker and military games)
• Cooperative games and markets (How to fairly divide a treasure)

 

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

 

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Homework 1. Basic technique

R. Weinstock

1. p. 44 exersis 2;
2. p. 44 exersis 3;
3. p. 45 exersis 4;
4. p. 46 exersis 7;
5. p. 47 exersis 13;
6. p. 47 exersis 14.

 

Homework 2. Isoperimentric system. Optimal control problems

R. Weinstock

1. p. 62 exersis 1;
2. p. 62 exersis 2;
3. p. 65 exersis 9;
4. p. 65 exersis 10;
5. p. 65 exersis 11.
6. Formulate the isoperimetric problem as a problem of optimal control. Solve.
7. Formulate and solve the following control problem:
8. A mass is initially (t=0) at rest (x(0)=x'(0)=0). What force $F=F(t)$ must be applied to the mass to move it as far as possible at the final time $t=1$ (x(1) -> max) if (1) the speed of the mass in the final time must be zero, (2) the absolute value of the force is less that one all the time: $ | F(t) | \leq 1 $, forall $ t \in [0, 1]$?

     

Class Notes (supplements to the textbook)