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The Action Principle and Partial Differential Equations. (AM-146), Volume 146 : Annals of Mathematics Studies, 146 - Demetrios Christodoulou

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

Annals of Mathematics Studies, 146

Paperback Published: 17th January 2000
ISBN: 9780691049571
Number Of Pages: 328

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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.

General Introductionp. 3
Introductionp. 7
The Lagrangian Picturep. 8
The Hamiltonian Picturep. 19
Examplesp. 28
Introductionp. 51
The Canonlicd and Symplectic Formsp. 57
Symplectic Transformationsp. 62
The Equations of Variationp. 79
The Circulation Theoremp. 84
The Euler Systemp. 87
Irrotational Solutionsp. 96
The Equation of Continuityp. 99
Introductionp. 105
Compatible Currentsp. 108
Null Currents and Null Lagragiansp. 125
The Source Equationsp. 128
The Generic Case n >1 & m >2p. 133
The Separable Case m >2p. 141
The Case m = 2p. 144
Lie Flows and the Noether Currentp. 146
Sections of Vector Bundlesp. 159
Introductionp. 191
Relative Lagrangiansp. 195
Ellipticity and Hyperbolicityp. 220
The Domain of Dependencep. 240
The Electromagnetic Fieldp. 263
Electromagnetic Symplectic Structurep. 272
Electromagnetic Compatible Currentsp. 282
Causality in Electromagnetic etic Theoryp. 299
Bibliographyp. 315
Indexp. 317
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691049571
ISBN-10: 0691049572
Series: Annals of Mathematics Studies, 146
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 328
Published: 17th January 2000
Country of Publication: US
Dimensions (cm): 23.5 x 15.88  x 1.91
Weight (kg): 0.48