Bryant R., Griffiths P., Grossman D., Exterior Differential Systems and Euler-Lagrange Partial Differential Equations.pdf

文档介绍：arXiv:math/0207039v1 [] 3 Jul 2002
ue-arnePrilDffrnilEquations Differential Partial Euler-Lagrange
oetByn hli rffih ailGrossman Daniel Griffiths Phillip Bryant Robert
xeirDffrnilSsesand Systems Differential Exterior
eray1 2008 1, February
ii
Contents
Preface v
Introduction vii
1 Lagrangians and Poincar´e-Cartan Forms 1
Lagrangians and Contact Geometry . . . . . . . . . . . . . . . . 1
The Euler-Lagrange System . . . . . . . . . . . . . . . . . . . . . 7
Variation of a Legendre Submanifold . . . . . . . . . . . . 7
Calculation of the Euler-Lagrange System . . . . . . . . . 8
The Inverse Problem . . . . . . . . . . . . . . . . . . . . . 10
Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Hypersurfaces in Euclidean Space . . . . . . . . . . . . . . . . . . 21
The Contact Manifold over En+1 . . . . . . . . . . . . . . 21
Euclidean-invariant Euler-Lagrange Systems . . . . . . . . 24
Conservation Laws for Minimal Hypersurfaces . . . . . . . 27
2 The Geometry of Poincar´e-Cartan Forms 37
The Equivalence Problem for n = 2 . . . . . . . . . . . . . . . . . 39
Neo-Classical Poincar´e-Cartan Forms . . . . . . . . . . . . . . . . 52
Digression on Affine Geometry of Hypersurfaces . . . . . . . . . . 58
The Equivalence Problem for n 3 . . . . . . . . . . . . . . . . . 65
≥
The Prescribed Mean Curvature System . . . . . . . . . . . . . . 74
3 Conformally Invariant Systems 79
Background Material on Conformal Geometry . . . . . . . . . . . 80
Flat Conformal Space . . . . . . . . . . . . . . . . . . . . 80
The Conformal Equivalence Problem . . . . . . . . . . . . 85
The Conformal Laplacian . . . . . . . . . . . . . . . . . . 93
Conformally Invariant Poincar´e-Cartan
Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
The Conformal Branch of the Equivalence Problem . . . . . . . .