Homework
MATH 7581
Date Chapter.Section/Topics or Page/Problems
T Jan 9 Lecture 1: Introduction: functionals
Exercises: 1 and 2
Th 11 Lecture 2: Weak extremals
(First variation to C^2 Euler-Lagrange Equation)
Exercises: 3-10
T 16 Lecture 3: The Euler-Lagrange equation
(Lemma of DuBois-Reymond to Exercise 20)
Exercises: 11-20
Th 18 snow
T 23 Lecture 4: Hamilton's action integral
(variational theory of motion)
Exercises: 21-22 (There may be some inconsistency
in these exercise numbers due to inserts from lectures
given below but incorporated earlier in the notes)
Th 25 Lecture 5: free boundary conditions
(new exercise 21 here)
Lecture 5: technique of the first integral
T 30 Lecture 6: C^2 inner variations
Exercises 22-23 (a new exercise 22)
Th Feb 1 exam 1
T 6 Lecture 7: Constraints; Lagrange multipliers
Th 8 project proposals due
Lecture 8: Second variation stability
T 13 Lecture 9: The brachistochrone problem
Th 15 Lecture 9: Convex functionals; sufficient conditions
T 20 Lecture 10: The hanging chain---extremals
Th 22 Lecture 10 (continued) The hanging chain---minimality subject to constraints
T 27 Lecture 10a Lagrangian constraints (see Lecture 4)
Th Mar 1 Lecture 10a (continued) Lagrangian constraints (continued)
The cantilevered (elastic) beam
T 6
Th 8 exam 2 (Preliminary project reports)
T 13
Th 15
T 20 spring break
Th 22 spring break
T 27 Beams and soap films and discontinuous potentials
Th 29
T Apr 3
Th 5 Introduction to the direct methods in the calculus of variations
T 10 Proof of the Arzela-Ascoli theorem
Th 12 Introduction to functional analysis for the direct methods
T 17 Dual spaces and weak convergence
hanging chain project
Th 19 last lecture
Th May 3 final exam 11:30-2:20
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M Jan 9 Introduction: Minimization of Functionals
(start reading Sagan Chapter 1)
W 11 Introduction (continued)
The Euler-Lagrange Equation (Indirect Methods); read Sagan 1.1-3.
Homework Assignment 1
M 16 holiday
W 18 The Euler-Lagrange Equation (continued)
M 23 Homework 1
Endpoint Conditions
Read Sagan Chapter 2 (through 2.6)
W 25 The First Integral and Noether's Theorem
M 30 Measuring Sets and Integration
Differentiability of monotone functions
W Feb 1 Overview of Integration Theory, L^p
M 6 More on L^p; Lebesgue points and differentiability
W 8 Homework 2
Introduction to constraints and Lagrange multipliers
M 13 Lagrange's and Mayer's Generalizations (Sagan Chapter 6)
W 15 Homeworks 3 and 4
M 20 Sufficient Conditions: Null Lagrangians and Fields of Extremals
W 22 Sufficient Conditions: Eikonals
M 27 Sufficient Conditions: Jacobi Fields
Homework 5
W 29 Homework; Rotational Minimal Surfaces; Fermat's Principle
M Mar 5 Homework; Newton's Optimal Shape; Drag
W 7 Homework 6
M 12 Project Proposals Due
Multidimensional Variational Problems
W 14 Newton's Optimal shape and more on Multivariable Variational Problems
M 19 Spring Break
W 21 Spring Break
M 26
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OLD SCHEDULE
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W 20 The first variation and the Euler-Lagrange equation (intro)
Go over Homework Assignment 1
M 25 The Euler-Lagrange equation; minimal surfaces of revolution
(Should be able to do Homework Assignment 1.5)
W 27 Normed Linear Spaces; R^n and C^k spaces
Parametric Variational Problems (intro)
Action Integrals and Newton's 2nd Law
(Should be able to do Homework Assignment 1.75)
M Feb 1 Go over Homework Assignments 1,1.5,1.75
What Euler-Lagrange equations can tell you; weak solutions
W 3 computing Euler-Lagrange equations (students 20 pts on exam)
M 8 Weak solutions of the equation y'= 0.
W 10 Homework problems and overview of real analysis, Sobolev space
M 15 Outline of material for the exam (be there or b^2)
W 17 Computing Euler-Lagrange Equations (students)
M 22 Midterm
W 24 Midterm Problem 3, Overview of Course Topics, &
Introduction to Variational Problems with Constraints (Sagan 6.1-6)
M March 1 Constraints (continued)
A little more computation of Euler-Lagrange Equations
Exams returned---go over/discuss exams
Project Proposals (due Monday March 15)
:HEADS UP: We're going to try to do some of chapter 5 (start reading)
W 3 Show me what you learned from the exam
(subtext: What you didn't learn before the exam)
M 8
W 10
M 15 Project Proposals Due (These projects should take you about
a month to 1.5 months, so start working.)
W 17 Intro to Optimal Control (Sagan 1.7 and 5.1)
M 22 Dynamic Programming (Sagan 5.2)
W 24 Hamilton-Jacobi Equation (Sagan 5.3)
M 29 Legendre Transform (5.3 continued)
W 31 Jacobi's Theorem (Sagan 5.4)
M April 5 Examples
W 7 Quasilinear First-Order PDE in two variables
M 12 Control Problems (Sagan 5.5-6)
W 14 More on Control (Sagan 5.7)
M 19 *
W 21 Pontryagin Minimum Principle (Sagan 5.7)
26 Project Presentations: Sumit Jain and Karthik Raveendran
28 Project Presentations; Galager Lecture on Mass Transport
M May 3 2:50-5:40 Final Exam Period