Mathematics 5500 Calculus of Variations
M-5500 Calculus of Variations
Winter 2012
M W F 2:00 - 2:50 JWB 333
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
Notes:
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
- 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
HW1
HW2
HW3
HW4
HW5