Action Principle and Partial Differential Equations

Demetrios Christodoulou is Professor of Mathematics at Princeton University. He has been awarded a John and Catherine MacArthur Fellowship, as well as a John Simon Guggenheim Fellowship. His previous book The Global Nonlinear Stability of the Minkowski Space (Princeton), cowritten with Sergiu Klainerman, won the Bücher Memorial Prize of the American Mathematical Society.

• General Introduction 3 1 1.0 Introduction 7 1.1 The Lagrangian Picture 8 1.2 The Hamiltonian Picture 19 1.3 Examples 28 2 2.0 Introduction 51 2.1 The Canonlicd and Symplectic Forms 57 2.2 Symplectic Transformations 62 2.3 The Equations of Variation 79 2.4 The Circulation Theorem 84 2.5 The Euler System 87 2.6 Irrotational Solutions 96 2.7 The Equation of Continuity 99 3 3.0 Introduction 105 3.1 Compatible Currents 108 3.2 Null Currents and Null Lagragians 125 3.3 The Source Equations 128 3.4 The Generic Case n > 1 & m > 2 133 3.5 The Separable Case m > 2 141 3.6 The Case m = 2 144 3.7 Lie Flows and the Noether Current 146 4 4.1 Sections of Vector Bundles 159 5 5.0 Introduction 191 5.1 Relative Lagrangians 195 5.2 Ellipticity and Hyperbolicity 220 5.3 The Domain of Dependence 240 6 6.1 The Electromagnetic Field 263 6.2 Electromagnetic Symplectic Structure 272 6.3 Electromagnetic Compatible Currents 282 6.4 Causality in Electromagnetic etic Theory 299 Bibliography 315 Index 317