M-5500 and 6880-001
Calculus of Variations

Spring 2018

Class meets: MW / 11:50AM-01:10PM LS 102
Office hours: W, 1:30-2:30 PM or by appointment

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



I will work on the notes and edit them during the semester.
  1. Introduction
  2. Stationarity condition 1. Euler equation
  3. Geometric optics, brachistochrone, minimal surface of revolution
  4.  Second Variation I (1d). Legendre, Weierstrass, Jacobi tests. Examples

  5. Constrained problems: Lagrange multipliers, Isoperimentric problems. Functionals
  6. Isoperimetric and geodesics problems
  7. Constraints and Hamiltonian. Lagrangian mechanics
  8. Legendre Duality: Dual Variational Principles
  9. Approximation with penalty
  10. Two body problem in celestial mechanics
  11. Numerical methods
  12. Irregular solutions: Sketch

  13. Reminder. Vector and matrix differentiation, Integral formulas
  14. Stationarity condition 2. Multiple integrals.One minimizer.
  15. Stationarity condition 3. Multiple integrals. Several minimizers. Examples: Elasticity, Complex conductivity
  16. Optimal design: Problems with differential constraints
  17. Second Variation 2 (Multivariable). Legendre, Weierstrass, Jacobi tests.

  18. Variation of Domain. Applications to geometry

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
  • Inequalities that Imply the Isoperimetric Inequality: an article by Andrejs Treibergs: http://www.math.utah.edu/~treiberg/isoperim/isop.pdf

     Homework (will be updated)

    HW1 2018
    HW2 2018 

    HW3 2018 
    HW4 2018 
    Sources for Numerical methods
    Shooting methods

    Approximation method (see also HW5)
    HW5 2018
    Ref for formulation of the control problems:
    1. From Calculus of Variations to Optimal Control
    by Daniel Liberzon
    University of Illinois at Urbana-Champaign

    2. An Introduction to Mathematical Optimal Control Theory
    by Lawrence C. Evans
    University of California, Berkeley

    Relaxation of nonconvex variational problems:


    Math 5500-001 Review Session:
     4/27/18 - 1:30PM - 3:30PM Scheduled LCB 121

    Problem for the final exam
    eturn your work at Monday, 4/29/18 before 5 pm.
    Notice: an extra-credit problem is added.

    Homework assignments from from the last year

    HW2 - approximates

    HW3 - constraints
    HW3 - the new file
    HW4 - numerical solutions
    HW5 - Hamiltonian and Legendre transform

    HW5 - duality
    NW6 - see the note
    HW 8 (PDE)

    Final HW 2017