Exterior Differential Systems and EulerLagrange Partial Differential Equations
216 pages

6 x 9

© 2003
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 PoincaréCartan 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, EulerLagrange PDE systems, and higherorder 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.
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.
Contents
Preface
Introduction
1. Lagrangians and PoincaréCartan Forms
1.1 Lagrangians and Contact Geometry
1.2 The EulerLagrange System
1.3 Noether's Theorem
1.4 Hypersurfaces in Euclidean Space
2. The Geometry of PoincaréCartan Forms
2.1 The Equivalence Problem for n = 2
2.2 NeoClassical PoincaréCartan 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 EulerLagrange Systems
3.1 Background Material on Conformal Geometry
3.2 Confromally Invariant PoincaréCartan Forms
3.3 The Conformal Branch of the Equivalence Problem
3.4 Conservation Laws for Du = Cu n+2/n2
3.5 Conservation Laws for Wave Equations
4. Additional Topics
4.1 The Second Variation
4.2 EulerLagrange PDE Systems
4.3 HigherOrder Conservation Laws
Introduction
1. Lagrangians and PoincaréCartan Forms
1.1 Lagrangians and Contact Geometry
1.2 The EulerLagrange System
1.3 Noether's Theorem
1.4 Hypersurfaces in Euclidean Space
2. The Geometry of PoincaréCartan Forms
2.1 The Equivalence Problem for n = 2
2.2 NeoClassical PoincaréCartan 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 EulerLagrange Systems
3.1 Background Material on Conformal Geometry
3.2 Confromally Invariant PoincaréCartan Forms
3.3 The Conformal Branch of the Equivalence Problem
3.4 Conservation Laws for Du = Cu n+2/n2
3.5 Conservation Laws for Wave Equations
4. Additional Topics
4.1 The Second Variation
4.2 EulerLagrange PDE Systems
4.3 HigherOrder Conservation Laws
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