Introduction to the Calculus of Variations

by Hans Sagan

Overview

". . . eminently suitable as a text for an introductory course: the style is pleasant; the prerequisites are kept to a minimum . . . and the pace of the development is appropriate for most students at the senior or first year graduate level." — American Mathematical Monthly
The purpose of this text is to lay a broad foundation for an understanding of the problems of the calculus of variations and its many methods and techniques, and to prepare readers for the study of modern optimal control theory. The treatment is limited to a thorough discussion of single-integral problems in one or more unknown functions, where the integral is employed in the riemannian sense.
The first three chapters deal with variational problems without constraints. Chapter 4 is a self-contained treatment of the homogeneous problem in the two-dimensional plane. In Chapter 5, the minimum principle of Pontryagin as it applies to optimal control problems of nonpredetermined duration, where the state variables satisfy an autonomous system of first-order equations, is developed to the extent possible by classical means within the general framework of the Hamilton-Jacobi theory. Chapter 6 is devoted to a derivation of the multiplier rule for the problem of Mayer with fixed and variable endpoints and its application to the problem of Lagrange and the isoperimetric problem. In the last chapter, Legendre's necessary condition for a weak relative minimum and a sufficient condition for a weak relative minimum are derived within the framework of the theory of the second variation.
This book, which includes many strategically placed problems and over 400 exercises, is directed to advanced undergraduate and graduate students with a background in advanced calculus and intermediate differential equations, and is adaptable to either a one- or two-semester course on the subject.

Product Details

ISBN-13:	9780486138022
Publisher:	Dover Publications
Publication date:	03/29/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	480
File size:	64 MB
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Table of Contents

PREFACE
ACKNOWLEDGMENTS
CHAPTER 1 EXTREME VALUES OF FUNCTIONALS
1.1 INTRODUCTION
1.2 FUNCTIONALS
1.3 NECESSARY CONDITIONS FOR RELATIVE EXTREME VALUES OF REAL-VALUED FUNCTIONS OF ONE REAL VARIABLE
1.4 NORMED LINEAR SPACEW
1.5 THE GÂTEAUX VARIATION OF A FUNCTIONAL
1.6 THE SPACE OF ADMISSIBLE VARIATIONS
1.7 FIRST NECESSARY CONDITION FOR A RELATIVE MINIMUM OF A FUNCTIONAL
1.8 THE SECOND GÂTEAUX VARIATION AND A SECOND NECESSARY CONDITION FOR A RELATIVE MINIMUM OF A FUNCTIONAL
BRIEF SUMMARY
APPENDIX
A1.9 RELATIVE EXTREME VALUES OF REAL-VALUED FUNCTIONS OF n REAL VARIABLES
CHAPTER 2 THE THEORY OF THE FIRST VARIATION
2.1 WEAK AND STRONG RELATIVE EXTREME VALUES
2.2 FIRST NECESSARY CONDITION FOR THE SIMPLEST VARIATIONAL PROBLEM
2.3 THE EULER-LAGRANGE EQUATION
2.4 LAGRANGE'S METHOD
2.5 DISCUSSION OF THE EULER-LAGRANGE EQUATION
2.6 THE PROBLEM OF MINIMAL SURFACES OF REVOLUTION
2.7 NATURAL BOUNDARY CONDITIONS
2.8 TRANSVERSALITY CONDITIONS
2.9 BROKEN EXTREMALS AND THE WEIRSTRASS-ERDMANN CORNER CONDITIONS
2.10 SMOOTHING OF CORNERS
2.11 GENERALIZATION TO MORE THAN ONE UNKNOWN FUNCTION
2.12 THE EULER-LAGRANGE EQUATIONS IN CANONICAL FORM
BRIEF SUMMARY
APPENDIX
A2.13 THE PROBLEM IN TWO UNKNOWN FUNCTIONS WITH VARIABLE ENDPOINTS
A2.14 INVARIANCE OF THE EULER-LAGRANGE EQUATIONS
A2.15 HAMILTON'S PRINCIPLE OF STATIONARY ACTION
A2.16 NOETHER'S INTEGRATION OF THE EULER-LAGRANGE EQUATION-CONSERVATION LAWS IN MECHANICS
A2.17 GENERALIZATION TO MORE THAN ONE INDEPENDENT VARIABLE
CHAPTER 3 THEORY OF FIELDS AND SUFFICIENT CONDITIONS FOR A STRONG RELATIVE EXTREMUM
3.1 FIELDS
3.2 HILBERT'S INVARIANT INTEGRAL
3.3 TRANSFORMATION OF THE TOTAL VARIATION
3.4 AN EXAMPLE OF A STRONG MINIMUM
3.5 FIELD CONSTRUCTION AND THE JACOBI EQUATION
3.6 THE ZEROS OF THE SOLUTIONS OF THE JACOBI EQUATION-CONJUGATE POINTS
3.7 CONJUGATE POINTS AND FIELD EXISTENCE
3.8 A SUFFICIENT CONDITION FOR A WEAK MINIMUM
3.9 A NECESSARY CONDITION FOR A STRONG RELATIVE MINIMUM
3.10 A SUFFICIENT CONDITION FOR THE PROBLEM IN n UNKNOWN FUNCTIONS
BRIEF SUMMARY
APPENDIX
A3.11 SUFFICIENT CONDITIONS FOR THE VARIABLE-ENDPOINT PROBLEM
A3.12 EXISTENCE OF A TRANSVERSAL FIELD
A3.13 FOCAL POINTS IN TRANSVERSAL FIELD
A3.14 "FIELD, INVARIANT INTEGRAL, AND EXCESS FUNCTION OF THE PROBLEM IN TWO INDEPENDENT VARIABLES"
CHAPTER 4 THE HOMOGENEOUS PROBLEM
4.1 PARAMETER INVARIANCE OF INTEGRAL
4.2 PROPERTIES OF HOMOGENEOUS FUNCTIONS
4.3 WEAK AND STRONG RELATIVE EXTREMA
4.4 THE EULER-LAGRANGE EQUATIONS FOR THE HOMOGENEOUS PROBLEM
4.5 DISCUSSION OF THE EULER-LAGRANGE EQUATIONS
4.6 TRANSVERSALITY CONDITION
4.7 CARATHEODORY'S INDICATRIX
4.8 INTEGRALS OF THE EULER-LAGRANGE EQUATIONS
4.9 FIELD AND EXCESS FUNCTION
4.10 STRONG AND WEAK EXTREMA
BRIEF SUMMARY
CHAPTER 5 THE HAMILTON-JACOBI THEORY AND THE MINIMUM PRINCIPLE OF PONTRYAGIN
5.1 A FUNDAMENTAL LEMMA OF CARATHÉODORY
5.2 DYNAMIC PROGRAMMING
5.3 THE HAMILTON-JACOBI EQUATION
5.4 SOLUTION OF THE HAMILTON-JACOBI EQUATION-JACOBI'S THEOREM
5.5 THE HAMILTON-JACOBI EQUATION AND FIELD EXISTENCE
5.6 A GENERAL MINIMUM-INTEGRAL CONTROL PROBLEM
5.7 THE MINIMUM PRINCIPLE OF PONTRYAGIN
BRIEF SUMMARY
APPENDIX
A5.8 THE TIME-OPTIMAL CONTROL PROBLEM
A5.9 A NONAUTONOMOUS TERMINAL CONTROL PROBLEM OF PREDETERMINED DURATION
A5.10 THE MINIMUM PRINCIPLE AS A SUFFICIENT CONDITION FOR LINEAR CONTROL PROBLEMS OF FIXED DURATION
A5.11 BANG-BANG CONTROLS
A5.12 A PROBLEM OF LAGRANGE AS AN OPTIMAL CONTROL PROBLEM
CHAPTER 6 THE PROBLEM OF LAGRANGE AND THE ISOPERIMETRIC PROBLEM
6.1 VARIATIONAL PROBLEMS WITH CONSTRAINTS
6.2 THE PROBLEM OF MAYER AND A FUNDAMENTAL THEOREM OF UNDETERMINED SYSTEMS
6.3 THE LAGRANGE MULTIPLIER RULE
6.4 DISCUSSION OF THE LAGRANGE MULTIPLIER RULE
6.5 THE ISOPERIMETRIC PROBLEM
6.6 DISCUSSION OF THE ISOPERIMETRIC PROBLEM
6.7 PROOF OF THE FUNDAMENTAL THEOREM OF UNDERDETERMINED SYSTEMS
6.8 THE MAYER PROBLEM WITH A VARIABLE ENDPOINT
6.9 TRANSVERSALITY CONDITIONS FOR THE LAGRANGE PROBLEM WITH A VARIABLE ENDPOINT
6.10 A SUFFICIENT CONDITION FOR THE LAGRANGE PROBLEM
BRIEF SUMMARY
APPENDIX
A6.11 ON THE AUGMENTATION OF A MATRIX
A6.12 A LAGRANGE PROBLEM WITH FINITE CONSTRAINTS
CHAPTER 7 THE THEORY OF THE SECOND VARIATION
7.1 NECESSARY AND SUFFICIENT CONDITIONS FOR A WEAK MINIMUM
7.2 LEGENDRE'S NECESSARY CONDITION
7.3 BLISS' SECONDARY VARIATIONAL PROBLEM AND JACOBI'S NECESSARY CONDITION
7.4 LEGENDRE'S TRANSFORMATION OF THE SECOND VARIATION
7.5 A SUFFICIENT CONDITION FOR A WEAK RELATIVE MINIMUM
7.6 SCHEMATIC REVIEW OF THE SIMPLEST VARIATIONAL PROBLEM
7.7 THE SECOND VARIATION OF FUNCTIONAL OF n VARIABLES
7.8 THE STRENGTHENED LEGENDRE CONDITION
7.9 CONJUGATE POINTS AND JACOBI'S NECESSARY CONDITION
BRIEF SUMMARY
APPENDIX
A7.10 THE LEGENDRE CONDITION FOR THE HOMOGENEOUS PROBLEM
A7.11 THE JACOBI CONDITION FOR THE HOMOGENEOUS PROBLEM
BIBLIOGRAPHY
INDEX

Editorial Reviews

Reprint of the McGraw-Hill edition of 1969 (in their International Series in Pure and Applied Mathematics). Annotation c. Book News, Inc., Portland, OR (booknews.com)

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