Exterior differential systems and Euler-Lagrange partial differential equations / Robert Bryant, Phillip Griffiths, and Daniel Grossman.

Format:

Book

Author/Creator:

Bryant, Robert L.

Contributor:

Griffiths, Phillip, 1938-

Grossman, Daniel Andrew, 1974-

Series:

Chicago lectures in mathematics

Chicago lectures in mathematics series

Language:

English

Subjects (All):

Exterior differential systems.

Lagrange equations.

Physical Description:

vii, 213 pages : illustrations ; 24 cm.

Place of Publication:

Chicago : University of Chicago Press, 2003.

Summary:

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.

Contents:

1 Lagrangians and Poincare-Cartan Forms 9

1.1 Lagrangians and Contact Geometry 9

1.2 The Euler-Lagrange System 15

1.2.1 Variation of a Legendre Submanifold 15

1.2.2 Calculation of the Euler-Lagrange System 16

1.2.3 The Inverse Problem 18

1.3 Noether's Theorem 22

1.4 Hypersurfaces in Euclidean Space 29

1.4.1 The Contact Manifold over E[superscript n+1] 29

1.4.2 Euclidean-invariant Euler-Lagrange Systems 32

1.4.3 Conservation Laws for Minimal Hypersurfaces 35

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 n [greater than or equal] 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

3.1.1 Flat Conformal Space 88

3.1.2 The Conformal Equivalence Problem 93

3.1.3 The Conformal Laplacian 101

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[superscript n+2 / n-2] 118

3.4.1 The Lie Algebra of Infinitesimal Symmetries 119

3.4.2 Calculation of Conservation Laws 121

3.5 Conservation Laws for Wave Equations 126

3.5.1 Energy Density 129

3.5.2 The Conformally Invariant Wave Equation 131

3.5.3 Energy in Three Space Dimensions 135

4.1 The Second Variation 141

4.1.1 A Formula for the Second Variation 141

4.1.2 Relative Conformal Geometry 144

4.1.3 Intrinsic Integration by Parts 147

4.1.4 Prescribed Mean Curvature, Revisited 148

4.1.5 Conditions for a Local Minimum 153

4.2 Euler-Lagrange PDE Systems 158

4.2.1 Multi-contact Geometry 159

4.2.2 Functionals on Submanifolds of Higher Codimension 163

4.2.3 The Betounes and Poincare-Cartan Forms 166

4.2.4 Harmonic Maps of Riemannian Manifolds 172

4.3 Higher-Order Conservation Laws 176

4.3.1 The Infinite Prolongation 176

4.3.2 Noether's Theorem 180

4.3.3 The K = -1 Surface System 190

4.3.4 Two Backlund Transformations 198.

Notes:

Includes bibliographical references (pages 207-209) and index.

ISBN:

0226077934

0226077942

OCLC:

51804819