Mathematics 5500 Calculus of Variations M-5500 Calculus of Variations

Spring 2015

M 03:05 -03:55 JWB 308,      W 3:05 - 17:00, 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.
  1. Introduction
  2. Stationarity condition 1. Euler equation
  3. Geometric optics, brachistochrone, minimal surface of revolution
  4. Approximation with penalty
  5. Constrained problems 1. Lagrange multiplyers, Isoperimentric problems. Functional - superposition of integrals
  6. Constraints and Hamiltonian. Lagrangean mechanics
  7. Legendre Duality: Dual Variational Principles
  8. Reminder. Vector and matrix differentiation, Integral formulas
  9. Stationarity condition 2. Multiple integrals.
  10. Stationarity condition 3. Multiple integrals. Several minimizers. Examples: Elasticity, Complex conductivity
  11. Optimal design: Problems with differential constraints
  12. Second Variation I (1d). Legendre, Weierstrass, Jacobi tests. Examples
  13. Second Variation 2 (Multivariable). Legendre, Weierstrass, Jacobi tests.
  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

    • 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