M-5500 and 6880-001
Calculus of Variations
Spring 2019

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:

I. Necessary conditions (4 weeks)
1. Introduction. Historic notes
2. Variational derivative, Euler equation, First Integrals. Examples: Smooth approximation, brachistochrone,
3. Development: variational boundary conditions, Weierstass-Erdmann and transversality  conditions,  Examples.
4. Second variation: Legendre, Jacobi, Weierstrass tests. Examples. Minimal action principle, Relativistic approach.
5. Ill-posed problems, necessary conditions for existence of a solution, regularization (examples: minimal surface of revolution, etc.

II. Constrained problem (3.5 weeks)
1. Isoperimetric problem. Examples: Dido problem, Equilibrium of a heavy chain
2. Lagrange function. Example: Geodesics
3. Hamiltonian, Duality, Dual variational principles. Examples: Mechanics
4. Invariants. Noether theorem. Example:  Celestial two body problem.

III. Numerical methods (1.5 weeks)
1. Iterative shooting method for boundary value problem.
2. Rayleigh–Ritz method and generalizations: splines.
3. Introduction to Galerkin method and weak solutions.

IV. Basics of control theory (3 weeks)
1. Formalism, Hamiltonian. Conjugate system. Examples
2. Pontryagin's maximum principle.
3. Existence. Convexification.

V. Multivariable variational problems  (4 weeks)
1. Euler Lagrange equation for scalar potential. Nonlinear conductivity, p-Laplacian.
2. Euler Lagrange equation for vector potential. Elasticity, Electromagnetism.
3. Second variation: Jacobi and Weierstrass tests. Rank-One convexity.
4. Variation of Domain. Applications to geometry
5. Some optimal design problems: Design of sources.
6. Notion of Quasiconvexity. Example

Text: Lecture Notes:(will be posted)

Note 1 First Variation
Note 2 Second Variation
Note 3 Immediate Applications: Mechanics, Approximationn with Penalty.
Note 4 Introduction to Lagrange multipliers
Note 5 Isoperimetric and Pointwise Constrained Problems
Note 6 Hamiltonian
Note 7 Legendre Transform
Note 8 Solutions with unbounded derivatives
Note 9-1 Convexity: Infinitely often oscillating solutions (one minimizer)
Note 9-2 Convexity: Infinitely often oscillating solutions (several minimizers)
Note 10 Introduction to control theory

Note 11 Euler equation, single potential
Note 12 Boundary terms
Note 13 Vector minimizer
Note 14 Variation of domains
Note 15 Design of external sources
Note 16 Design of conducting composites

I will work on the notes and edit them during the semester.

Recommended reading

Two body problem:
http://web.mit.edu/8.01t/www/materials/modules/guide17.pdf

History of isoperimetric problem
https://www.maa.org/press/periodicals/convergence/the-sagacity-of-circles-a-history-of-the-isoperimetric-problem-introduction

Four hinges
https://www.maa.org/press/periodicals/convergence/the-sagacity-of-circles-a-history-of-the-isoperimetric-problem-the-work-of-jakob-steiner


  • I. M. Gelfand, S. V. Fomin Calculus of Variations Dover Publications, 2000
  • Lecture Notes by Erich Miersemann http://www.math.uni-leipzig.de/~miersemann/variabook.pdf
  • Introduction to the Modern Calculus of Variations by Filip Rindler https://warwick.ac.uk/fac/sci/maths/people/staff/filip_rindler/cov_ln.pdf
  • Robert Weinstock. Calculus of Variations with Applications to Physics and Engineering. Dover Publications, 1
  • Inequalities that Imply the Isoperimetric Inequality: an article by Andrejs Treibergs:

     Homework (will be updated)

    • HW1 Due January 23.
    HW2 Due February 11.
    HW3 Due February 25

    HW4     Due March 18.
    HW5
    HW6


    Final exam

    Grades are based on Homework (60%) and Final exam (40%)

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