calculus of variations & solution manual - russak

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

2、...................52Examples,Notation102.1Notation&Conventions.............................132.2ShortestDistances................................143FirstResults213.1TwoImportantAuxiliaryFormulas:.......................223.2TwoImportantAuxiliaryFormulasintheGeneralCase............26

3、4VariableEnd-PointProblems364.1TheGeneralProblem...............................384.2Appendix.....................................415HigherDimensionalProblemsandAnotherProofoftheSecondEulerEquation465.1VariationalProblemswithConstraints.....................475.1.1IsoparametricProblem

4、s..........................475.1.2PointConstraints..............................516IntegralsInvolvingMoreThanOneIndependentVariable597ExamplesofNumericalTechniques637.1IndirectMethods.................................637.1.1FixedEndPoints..............................637.1.2VariableE

5、ndPoints............................717.2DirectMethods..................................748TheRayleigh-RitzMethod828.1Euler’sMethodofFiniteDifferences.......................849Hamilton’sPrinciple9010DegreesofFreedom-GeneralizedCoordinates9711IntegralsInvolvingHigherDerivatives10412Pi

6、ecewiseSmoothArcsandAdditionalResults11013FieldTheoryJacobi’sNeccesaryConditionandSufficiency116iListofFigures1NeighborhoodSofX0..............................22NeighborhoodSofX0andaparticulardirectionH..............23TwodimensionalneighborhoodofX0showingtangentatthatpoint.....54Thecon

7、straintφ.................................65Thesurfaceofrevolutionforthesoapexample.................116Brachistochroneproblem.............................127AnarcconnectingX1andX2..........................158Admissiblefunctionηvanishingatendpoints(bottom)andvariousadmissiblefunctions

8、(top)..............