calculus of variations & solution manual - russak

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1、CALCULUSOFVARIATIONSMA4311LECTURENOTESI.B.RussakDepartmentofMathematicsNavalPostgraduateSchoolCodeMA/RuMonterey,California93943July9,2002c1996-ProfessorI.B.Russak1Contents1FunctionsofnVariables11.1UnconstrainedMinimum.............................11.2Const

2、rainedMinimization............................52Examples,Notation102.1Notation&Conventions.............................132.2ShortestDistances................................143FirstResults213.1TwoImportantAuxiliaryFormulas:.......................223.2TwoIm

3、portantAuxiliaryFormulasintheGeneralCase............264VariableEnd-PointProblems364.1TheGeneralProblem...............................384.2Appendix.....................................415HigherDimensionalProblemsandAnotherProofoftheSecondEulerEquation465.1V

4、ariationalProblemswithConstraints.....................475.1.1IsoparametricProblems..........................475.1.2PointConstraints..............................516IntegralsInvolvingMoreThanOneIndependentVariable597ExamplesofNumericalTechniques637.1Indirec

5、tMethods.................................637.1.1FixedEndPoints..............................637.1.2VariableEndPoints............................717.2DirectMethods..................................748TheRayleigh-RitzMethod828.1Euler’sMethodofFiniteDifference

6、s.......................849Hamilton’sPrinciple9010DegreesofFreedom-GeneralizedCoordinates9711IntegralsInvolvingHigherDerivatives10412PiecewiseSmoothArcsandAdditionalResults11013FieldTheoryJacobi’sNeccesaryConditionandSufficiency116iListofFigures1Neighborhood

7、SofX0..............................22NeighborhoodSofX0andaparticulardirectionH..............23TwodimensionalneighborhoodofX0showingtangentatthatpoint.....54Theconstraintφ.................................65Thesurfaceofrevolutionforthesoapexample............

8、.....116Brachistochroneproblem.............................127AnarcconnectingX1andX2..........................158Admissiblefunctionηvanishingatendpoints(bottom)andvariousadmissiblefunctions(top)..............