VariationalMethodswithApplicationsinScienceandEngineering

VariationalMethodswithApplicationsinScienceandEngineering
1Preliminaries31.1ABitofHistory41.2Introduction71.3Motivation81.3.1Optics81.3.2ShapeofaLiquidDrop101.3.3OptimizationofaRiver-CrossingTrajectory121.3.4Summary141.4ExtremaofFunctions141.5ConstrainedExtremaandLagrangeMultipliers171.6IntegrationbyParts201.7FundamentalLemmaoftheCalculusofVariations211.8AdjointandSelf-AdjointDifferentialOperators22Exercises262CalculusofVariations282.1FunctionalsofOneIndependentVariable292.1.1FunctionalDerivative302.1.2DerivationofEuler’sEquation312.1.3VariationalNotation332.1.4SpecialCasesofEuler’sEquation372.2NaturalBoundaryConditions442.3VariableEndPoints532.4Higher-OrderDerivatives562.5FunctionalsofTwoIndependentVariables562.5.1Euler’sEquation572.5.2MinimalSurfaces612.5.3DirichletProblem62viiviiiContents2.6FunctionalsofTwoDependentVariables642.7ConstrainedFunctionals662.7.1IntegralConstraints662.7.2Sturm-LiouvilleProblems742.7.3AlgebraicandDifferentialConstraints762.8SummaryofEulerEquations80Exercises813Rayleigh-Ritz,Galerkin,andFinite-ElementMethods903.1Rayleigh-RitzMethod913.1.1BasicProcedure913.1.2Self-AdjointDifferentialOperators943.1.3EstimatingEigenvaluesofDifferentialOperators963.2GalerkinMethod1003.3Finite-ElementMethods1033.3.1Rayleigh-Ritz–BasedFinite-ElementMethod1043.3.2Finite-ElementMethodsinMultidimensions109Exercises110PARTIIPHYSICALAPPLICATIONS1154Hamilton’sPrinciple1174.1Hamilton’sPrincipleforDiscreteSystems1184.2Hamilton’sPrincipleforContinuousSystems1284.3Euler-LagrangeEquations1314.4InvarianceoftheEuler-LagrangeEquations1364.5DerivationofHamilton’sPrinciplefromtheFirstLawofThermodynamics1374.6ConservationofMechanicalEnergyandtheHamiltonian1414.7Noether’sTheorem–ConnectionBetweenConservationLawsandSymmetriesinHamil 本软件ID:9427941