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Mathematics 5500 Calculus of Variations
M-5500 Calculus of Variations
Spring 2014
M 3:05-03:55 PM JWB 30, W 3:05 -500 pm JTB 110
Office: JWB 225
Telephone: 581-6822
E-mail: cherk@math.utah.edu
Every problem of the calculus of variations has a solution,
provided that the word `solution' is suitably
understood.
David Hilbert
Syllabus
Notes:
I will work on the notes and edit them during the semester. Be
aware that the text might vary.
Preliminaries (USAG talk at November 2013)
- Introduction
- Stationarity condition 1. Euler equation
-
Geometric optics, brachistochrone, minimal
surface of revolution
- > Approximation with penalty
- Reminder. Vector and matrix
differentiation, Interal formulas
- Stationarity condition 2. Multiple
integrals.
- Stationarity condition 3. Multiple
integrals. Several minimizers. Examples: Elasticity, Complex
conductrivity
- Second Variation I (1d). Legendre,
Weierstrass, Jacobi tests. Examples
- Second Variation 2 (Multivariable).
Legendre, Weierstrass, Jacobi tests.
- Constrained problems 1. Lagrange
multiplyers, Isoperimentric problems. Functional -
superposition of integrals
- Constraints and Hamiltonian. Lagrangean
mechanics
- Legendre Duality: Dual Variational
Principles
- Optimal design: Problems with diffrential
constraints
- Irregular solutions: Sketch
Recommended reading
Robert Weinstock. Calculus of Variations with Applications to
Physics and Engineering. Dover Publications, 1974.
I. M. Gelfand, S. V. Fomin Calculus of Variations Dover
Publications, 2000
Wikipedia
Homework
HW1
HW2