Introduction to the Calculus of Variations

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