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