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

*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