Selected Chapters in the Calculus of VariationsPDF电子书下载

0.1 Introduction 1

0.2 On these lecture notes 2

1 One-dimensional variational problems 3

1.1 Regularity of the minimals 3

1.2 Examples 9

1.3 The accessory variational problem 16

1.4 Extremal fields for n=1 20

1.5 The Hamiltonian formulation 25

1.6 Exercises to Chapter 1 30

2 Extremal fields and global minimals 33

2.1 Global extremal fields 33

2.2 An existence theorem 36

2.3 Properties of global minimals 42

2.4 A priori estimates and a compactness property 50

2.5 Mα for irrational α，Mather sets 56

2.6 Mα for rational α 73

2.7 Exercises to chapter Ⅱ 81

3 Discrete Systems，Applications 83

3.1 Monotone twist maps 83

3.2 A discrete variational problem 94

3.3 Three examples 98

3.4 A second variational problem 104

3.5 Minimal geodesics on T2 105

3.6 Hedlund’s metric on T3 108

3.7 Exercises to chapter Ⅲ 113

Bibliography 115

A Remarks on the literature 117

Additional Bibliography 121

Index 129