Exterior Differential Systems & Euler-Lagrange Partial Differential Equations

Preface vii Introduction 1(8) Lagrangians and Poincare-Cartan Forms 9(36) Lagrangians and Contact Geometry 9(6) The Euler-Lagrange System 15(7) Variation of a Legendre Submanifold 15(1) Calculation of the Euler-Lagrange System 16(2) The Inverse Problem 18(4) Noether's Theorem 22(7) Hypersurfaces in Euclidean Space 29(16) The Contact Manifold over En+1 29(3) Euclidean-invariant Euler-Lagrange Systems 32(3) Conservation Laws for Minimal Hypersurfaces 35(10) The Geometry of Poincare-Cartan Forms 45(42) The Equivalence Problem for n = 2 47(13) Neo-Classical Poincare-Cartan Forms 60(6) Digression on Affine Geometry of Hypersurfaces 66(7) The Equivalence Problem for n ≥ 3 73(9) The Prescribed Mean Curvature System 82(5) Conformally Invariant Euler-Lagrange Systems 87(54) Background Material on Conformal Geometry 88(17) Flat Conformal Space 88(5) The Conformal Equivalence Problem 93(8) The Conformal Laplacian 101(4) Conformally Invariant Poincare-Cartan Forms 105(5) The Conformal Branch of the Equivalence Problem 110(8) Conservation Laws for Δu = Cun+2/n-2 118(8) The Lie Algebra of Infinitesimal Symmetries 119(2) Calculation of Conservation Laws 121(5) Conservation Laws for Wave Equations 126(15) Energy Density 129(2) The Conformally Invariant Wave Equation 131(4) Energy in Three Space Dimensions 135(6) Additional Topics 141(1) The Second Variation 141(1) A Formula for the Second Variation 141(3) Relative Conformal Geometry 144(3) Intrinsic Integration by Parts 147(1) Prescribed Mean Curvature, Revisited 148(5) Conditions for a Local Minimum 153(5) Euler-Lagrange PDE Systems 158(1) Multi-contact Geometry 159(4) Functionals on Submanifolds of Higher Codimension 163(3) The Betounes and Poincare-Cartan Forms 166(6) Harmonic Maps of Riemannian Manifolds 172(4) Higher-Order Conservation Laws 176(1) The Infinite Prolongation 176(4) Noether's Theorem 180(10) The K = --1 Surface System 190(8) Two Backlund Transformations 198