Mathematics 5500 Calculus of Variations M-5500 Calculus of Variations

Spring 2014   MWF 03:05 PM-03:55 PM JWB 308

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

Preliminaries: Search for extremals in nature (USAG talk November 2013)

Notes:

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