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Other titles in the Chicago Lectures in Mathematics series:
Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Chicago Lectures in Mathematics)by Phillip Griffiths
Synopses & Reviews
In Exterior Differential Systems, the authors present the results of their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincaré-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.
About 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.
Table of Contents
1. Lagrangians and Poincaré-Cartan Forms
1.1 Lagrangians and Contact Geometry
1.2 The Euler-Lagrange System
1.3 Noether's Theorem
1.4 Hypersurfaces in Euclidean Space
2. The Geometry of Poincaré-Cartan Forms
2.1 The Equivalence Problem for n = 2
2.2 Neo-Classical Poincaré-Cartan Forms
2.3 Digression on Affine Geometry for Hypersurfaces
2.4 The Equivalence Problem for n > 3
2.5 The Prescribed Mean Curvature System
3. Conformally Invariant Euler-Lagrange Systems
3.1 Background Material on Conformal Geometry
3.2 Confromally Invariant Poincaré-Cartan Forms
3.3 The Conformal Branch of the Equivalence Problem
3.4 Conservation Laws for Du = Cu n+2/n-2
3.5 Conservation Laws for Wave Equations
4. Additional Topics
4.1 The Second Variation
4.2 Euler-Lagrange PDE Systems
4.3 Higher-Order Conservation Laws
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