Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

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Overview

In Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, the authors further their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincare-Cartan forms. They also cover certain aspects of the theory of exterior differential systems, which provides the language and techniques for the entire study. Because it plays a central role in uncovering geometric properties of differential equations, the method of equivalence is particularly emphasized. In addition, the authors discuss conformally invariant systems at length, including results on the classification and application of symmetries and conservation laws. The book also covers the second variation, Euler-Lagrange PDE systems, and higher-order conservation laws. This timely synthesis of partial differential equations and differential geometry will be of fundamental importance to both students and experienced researchers working in geometric analysis.
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Product Details

  • ISBN-13: 9780226077949
  • Publisher: University of Chicago Press
  • Publication date: 7/28/2003
  • Series: Chicago Lectures in Mathematics Series
  • Edition description: 1
  • Pages: 216
  • Product dimensions: 6.00 (w) x 9.00 (h) x 0.50 (d)

Meet the Author

Robert Bryant is the J. M. Kreps Professor in the Department of Mathematics at Duke University.

Phillip Griffiths is the director of the Institute for Advanced Study and professor in the Department of Mathematics at Duke University.

Daniel Grossman was an L. E. Dickson Instructor in the Department of Mathematics at the University of Chicago at the time of writing, and is now a consultant at the Chicago office of the Boston Consulting Group.

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Table of Contents

Preface
Introduction
1 Lagrangians and Poincare-Cartan Forms 9
1.1 Lagrangians and Contact Geometry 9
1.2 The Euler-Lagrange System 15
1.3 Noether's Theorem 22
1.4 Hypersurfaces in Euclidean Space 29
2 The Geometry of Poincare-Cartan Forms 45
2.1 The Equivalence Problem for n = 2 47
2.2 Neo-Classical Poincare-Cartan Forms 60
2.3 Digression on Affine Geometry of Hypersurfaces 66
2.4 The Equivalence Problem for [actual symbol not reproducible] 3 73
2.5 The Prescribed Mean Curvature System 82
3 Conformally Invariant Euler-Lagrange Systems 87
3.1 Background Material on Conformal Geometry 88
3.2 Conformally Invariant Poincare-Cartan Forms 105
3.3 The Conformal Branch of the Equivalence Problem 110
3.4 Conservation Laws for [Delta]u = Cu [actual symbol not reproducible] 118
3.5 Conservation Laws for Wave Equations 126
4 Additional Topics 141
4.1 The Second Variation 141
4.2 Euler-Lagrange PDE Systems 158
4.3 Higher-Order Conservation Laws 176
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