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

Syllabus

Notes:

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
    https://en.wikipedia.org/wiki/Shooting_method
    Lecture
    https://www.mathworks.com/matlabcentral/fileexchange/32451-shooting-method?focused=5194030&tab=function&s_tid=gn_loc_drop
    Matlab
    https://www.mathworks.com/matlabcentral/fileexchange/32451-shooting-method?focused=5194030&tab=function&s_tid=gn_loc_drop
    Matematica
    https://www.mathworks.com/matlabcentral/fileexchange/32451-shooting-method?focused=5194030&tab=function&s_tid=gn_loc_drop

    Approximation method (see also HW5)
    https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Ritz_method
    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
    http://liberzon.csl.illinois.edu/teaching/cvoc/node45.html

    2. An Introduction to Mathematical Optimal Control Theory
    by Lawrence C. Evans
    University of California, Berkeley
    math.berkeley.edu/~evans/control.course.pdf
    ----
    Relaxation of nonconvex variational problems:
    http://www.math.utah.edu/~cherk/teach/12calcvar/150existance.pdf

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

    Problem for the final exam
    R    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

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


    Final HW 2017