5720 Introduction to Applied Math II:
Calculus of Variations, Equations of Physics, and Games
Addressed to graduate and senior undergraduate students in math,
science, and engineering. Grade is based on several homework assignments
and (optionally) a course project.
Course's web page: www.math.utah.edu/~cherk/teach/appl-math2002.html
Text: (i)
Robert Weinstock. Calculus of Variations with Applications to Physics and
Engineering. Dover, 1974. (ii) Notes.
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 curve or surface is "the best?" or: Why there is no "best" solution at all?
Extremal 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.
The Game theory formulizes another basic human passion: Gambling
and decision making. The theory, originated in 30-s by Von Neumann and
Morgenstern, is now widely used in economics, biology, political and military
science.
Preliminary Sillabus:
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Convexity and some classical problems
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Euler equations, Legendre and Weierstrass tests,
Variation of the interval and boundary conditions, Restricted problems.
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Hamiltonian mechanics: Symmetries and invariants
Duality and Legendre-Young transform.
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Multivariable problems
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Direct methods and Finite elements.
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Variational principles (Do we live
in an "optimal" world?)
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Ill-posed variational problems: "Solving" problems without solutions
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Introduction to Games and minimax problems