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.
Author(s): Christodoulou D.
Series: Annals of mathematics studies, 146
Publisher: Princeton University Press
Year: 2000
Language: English
Pages: 322
City: Princeton, NJ
Contents ......Page 6
General Introduction ......Page 10
1.0 Introduction ......Page 14
1.1 The Lagrangian Picture ......Page 15
1.2 The Hamiltonian Picture ......Page 26
1.3 Examples ......Page 35
2.0 Introduction ......Page 58
2.1 The Canonical and Symplectic Forms ......Page 64
2.2 Symplectic Transformations ......Page 69
2.3 The Equations of Variation ......Page 86
2.4 The Circulation Theorem ......Page 91
2.5 The Euler System ......Page 94
2.6 Irrotational Solutions ......Page 103
2.7 The Equation of Continuity ......Page 106
3.0 Introduction ......Page 112
3.1 Compatible Currents ......Page 115
3.2 Null Currents amd Null Lagragians ......Page 132
3.3 The Source Equations ......Page 135
3.4 The Generic Case n > 1 & m > 2 ......Page 140
3.5 The Separable Case m > 2 ......Page 148
3.6 The Case m = 2 ......Page 151
3.7 Lie Flows and the Noether Current ......Page 153
4.1 Sections of Vector Bundles ......Page 166
5.0 Introduction ......Page 198
5.1 Relative Lagrangians ......Page 202
5.2 Ellipticity and Hyperbolicity ......Page 227
5.3 The Domain of Dependence ......Page 247
6.1 The Electromagnetic Field ......Page 270
6.2 Electromagnetic Symplectic Structure ......Page 279
6.3 Electromagnetic Compatible Currents ......Page 289
6.4 Causality in Electromagnetic Theory ......Page 306
Bibliography ......Page 322
Index ......Page 324
