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 ==================== OLD SCHEDULES BELOW ==================== 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 ==================== OLD SCHEDULE ==================== 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