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Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

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.

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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