The Action Principle and Partial Differential Equations

This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

Demetrios Christodoulou

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

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