Mathematics 5500 Calculus of Variations
M-55 00
Calculus of Variations
Spring 2015
M 03:05 -03:55 JWB 308, W 3:05 -
17:00, 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.
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.
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
HW4
- multiple integrals
HW5 - duality