The Action Principle and Partial Differential Equations. (AM-146), Volume 146

✍ Scribed by Demetrios Christodoulou

• Publisher: Princeton University Press
• Year: 2016
• Tongue: English
• Leaves: 327
• Series: Annals of Mathematics Studies; 146
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
General Introduction
1.
1.0 Introduction
1.1 The Lagrangian Picture
1.2 The Hamiltonian Picture
1.3 Examples
2.
2.0 Introduction
2.1 The Canonical and Symplectic Forms
2.2 Symplectic Transformations
2.3 The Equations of Variation
2.4 The Circulation Theorem
2.5 The Euler System
2.6 Irrotational Solutions
2.7 The Equation of Continuity
3.
3.0 Introduction
3.1 Compatible Currents
3.2 Null Currents and Null Lagragians
3.3 The Source Equations
3.4 The Generic Case n>1 & m>2
3.5 The Separable Case m>2
3.6 The Case m = 2
3.7 Lie Flows and the Noether Current
4.
4.1 Sections of Vector Bundles
5.
5.0 Introduction
5.1 Relative Lagrangians
5.2 Ellipticity and Hyperbolicity
5.3 The Domain of Dependence
6.
6.1 The Electromagnetic Field
6.2 Electromagnetic Symplectic Structure
6.3 Electromagnetic Compatible Currents
6.4 Causality in Electromagnetic Theory
Bibliography
Index