Mathematics 5500 Calculus of Variations
M-55 00
Calculus of Variations
Spring 2017
MW / 11:50AM-01:10PM LS 102
Office hours: W, 1:30-2:30 PM or by appointment
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.
Introduction
Stationarity condition 1. Euler equation
Geometric optics, brachistochrone,
minimal surface of revolution
Approximation with penalty
Constrained problems 1. Lagrange
multiplyers, Isoperimentric problems. Functional -
superposition of integrals
Constraints and Hamiltonian. Lagrangean
mechanics
Legendre Duality: Dual Variational
Principles
Reminder. Vector and matrix
differentiation, Integral formulas
Stationarity condition 2. Multiple
integrals.
Stationarity condition 3. Multiple
integrals. Several minimizers. Examples: Elasticity,
Complex conductivity
Optimal design: Problems with
differential constraints
Second Variation I (1d). Legendre,
Weierstrass, Jacobi tests. Examples
Second Variation 2 (Multivariable).
Legendre, Weierstrass, Jacobi tests.
Variation of Domains
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
Inequalities
that Imply the Isoperimetric Inequality : an article by
Andrejs Treibergs:
http://www.math.utah.edu/~treiberg/isoperim/isop.pdf
Homework
HW1
HW2 - approximates
HW3
- constraints
HW3
- the new file
HW4
- numerical solutions
HW5
- Hamiltonian and Legendre transform
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HW5
- duality
NW6 - see the note
HW 8 (PDE)
Final HW 2017