<head> <meta http-equiv="content-type" content="text/html; charset=windows-1252"> 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

Instructor Andrej Cherkaev

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)
  1. Introduction
  2. Stationarity condition 1. Euler equation
  3. Geometric optics, brachistochrone, minimal surface of revolution
  4. > Approximation with penalty
  5. Reminder. Vector and matrix differentiation, Interal formulas
  6. Stationarity condition 2. Multiple integrals.
  7. Stationarity condition 3. Multiple integrals. Several minimizers. Examples: Elasticity, Complex conductrivity
  8. Second Variation I (1d). Legendre, Weierstrass, Jacobi tests. Examples
  9. Second Variation 2 (Multivariable). Legendre, Weierstrass, Jacobi tests.
  10. Constrained problems 1. Lagrange multiplyers, Isoperimentric problems. Functional - superposition of integrals
  11. Constraints and Hamiltonian. Lagrangean mechanics
  12. Legendre Duality: Dual Variational Principles
  13. Optimal design: Problems with diffrential constraints
  14. 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