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
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.
- Introduction
- Stationarity condition 1. Euler equation
- Geometric optics, brachistochrone,
minimal surface of revolution
- Second Variation I (1d).
Legendre, Weierstrass, Jacobi tests. Examples
- Constrained problems: Lagrange
multipliers, Isoperimentric problems. Functionals
- Isoperimetric
and geodesics problems
- Constraints and Hamiltonian. Lagrangian
mechanics
- Legendre Duality: Dual Variational
Principles
- Approximation with penalty
- Two
body problem in celestial mechanics
- Numerical methods
- Irregular solutions: Sketch
- Reminder. Vector and matrix
differentiation, Integral formulas
- Stationarity condition 2. Multiple
integrals.One minimizer.
- Stationarity condition 3. Multiple
integrals. Several minimizers. Examples: Elasticity,
Complex conductivity
- Optimal design: Problems with
differential constraints
- Second Variation 2 (Multivariable).
Legendre, Weierstrass, Jacobi tests.
- 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
Return 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