exterior diff systems and euler-lagrange partial diff eqns - bryant, griffiths and grossman

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1、ExteriorDierentialSystemsandEuler-LagrangePartialDierentialEquationsRobertBryantPhillipGrithsDanielGrossmanJuly3,2002arXiv:math.DG/0207039v13Jul2002iiContentsPrefacevIntroductionvii1LagrangiansandPoincare-CartanForms11.1LagrangiansandContactGeometry................11.2TheE

2、uler-LagrangeSystem.....................71.2.1VariationofaLegendreSubmanifold............71.2.2CalculationoftheEuler-LagrangeSystem.........81.2.3TheInverseProblem.....................101.3Noether'sTheorem..........................141.4HypersurfacesinEuclideanSpace............

3、......211.4.1TheContactManifoldoverEn+1..............211.4.2Euclidean-invariantEuler-LagrangeSystems........241.4.3ConservationLawsforMinimalHypersurfaces.......272TheGeometryofPoincare-CartanForms372.1TheEquivalenceProblemforn=2.................392.2Neo-ClassicalPoincare-Ca

4、rtanForms................522.3DigressiononAneGeometryofHypersurfaces..........582.4TheEquivalenceProblemforn3.................652.5ThePrescribedMeanCurvatureSystem..............743ConformallyInvariantSystems793.1BackgroundMaterialonConformalGeometry...........803.1.1FlatConf

5、ormalSpace....................803.1.2TheConformalEquivalenceProblem............853.1.3TheConformalLaplacian..................933.2ConformallyInvariantPoincare-CartanForms.................................973.3TheConformalBranchoftheEquivalenceProblem........102n+23.4Conservati

6、onLawsfor
u=Cun2................1103.4.1TheLieAlgebraofInnitesimalSymmetries.......1113.4.2CalculationofConservationLaws..............114iiiivCONTENTS3.5ConservationLawsforWaveEquations...............1183.5.1EnergyDensity........................1223.5.2TheConformallyInvarian

7、tWaveEquation........1233.5.3EnergyinThreeSpaceDimensions.............1274AdditionalTopics1334.1TheSecondVariation........................1334.1.1AFormulafortheSecondVariation............1334.1.2RelativeConformalGeometry................1364.1.3IntrinsicIntegrationbyParts......

8、..........1394.1.4PrescribedMeanCurvature,Revisited..........