Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

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University of Chicago Press


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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.

Product Details

ISBN-13: 9780226077932
Publisher: University of Chicago Press
Publication date: 07/28/2003
Series: Chicago Lectures in Mathematics Series
Edition description: 1
Pages: 216
Product dimensions: 6.20(w) x 9.20(h) x 0.70(d)

Table of Contents

1Lagrangians and Poincare-Cartan Forms9
1.1Lagrangians and Contact Geometry9
1.2The Euler-Lagrange System15
1.2.1Variation of a Legendre Submanifold15
1.2.2Calculation of the Euler-Lagrange System16
1.2.3The Inverse Problem18
1.3Noether's Theorem22
1.4Hypersurfaces in Euclidean Space29
1.4.1The Contact Manifold over E[superscript n+1]29
1.4.2Euclidean-invariant Euler-Lagrange Systems32
1.4.3Conservation Laws for Minimal Hypersurfaces35
2The Geometry of Poincare-Cartan Forms45
2.1The Equivalence Problem for n = 247
2.2Neo-Classical Poincare-Cartan Forms60
2.3Digression on Affine Geometry of Hypersurfaces66
2.4The Equivalence Problem for n [greater than or equal] 373
2.5The Prescribed Mean Curvature System82
3Conformally Invariant Euler-Lagrange Systems87
3.1Background Material on Conformal Geometry88
3.1.1Flat Conformal Space88
3.1.2The Conformal Equivalence Problem93
3.1.3The Conformal Laplacian101
3.2Conformally Invariant Poincare-Cartan Forms105
3.3The Conformal Branch of the Equivalence Problem110
3.4Conservation Laws for [Delta]u = Cu[superscript n+2 / n-2]118
3.4.1The Lie Algebra of Infinitesimal Symmetries119
3.4.2Calculation of Conservation Laws121
3.5Conservation Laws for Wave Equations126
3.5.1Energy Density129
3.5.2The Conformally Invariant Wave Equation131
3.5.3Energy in Three Space Dimensions135
4Additional Topics141
4.1The Second Variation141
4.1.1A Formula for the Second Variation141
4.1.2Relative Conformal Geometry144
4.1.3Intrinsic Integration by Parts147
4.1.4Prescribed Mean Curvature, Revisited148
4.1.5Conditions for a Local Minimum153
4.2Euler-Lagrange PDE Systems158
4.2.1Multi-contact Geometry159
4.2.2Functionals on Submanifolds of Higher Codimension163
4.2.3The Betounes and Poincare-Cartan Forms166
4.2.4Harmonic Maps of Riemannian Manifolds172
4.3Higher-Order Conservation Laws176
4.3.1The Infinite Prolongation176
4.3.2Noether's Theorem180
4.3.3The K = -1 Surface System190
4.3.4Two Backlund Transformations198

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