calculus of variations & solution manual - russak

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

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

3、eneralCase............264VariableEnd-PointProblems364.1TheGeneralProblem...............................384.2Appendix.....................................415HigherDimensionalProblemsandAnotherProofoftheSecondEulerEquation465.1VariationalProblemswithConstraints............

4、.........475.1.1IsoparametricProblems..........................475.1.2PointConstraints..............................516IntegralsInvolvingMoreThanOneIndependentVariable597ExamplesofNumericalTechniques637.1IndirectMethods.................................637.1.1FixedEndPoin

5、ts..............................637.1.2VariableEndPoints............................717.2DirectMethods..................................748TheRayleigh-RitzMethod828.1Euler’sMethodofFiniteDifferences.......................849Hamilton’sPrinciple9010DegreesofFreedom-Generali

6、zedCoordinates9711IntegralsInvolvingHigherDerivatives10412PiecewiseSmoothArcsandAdditionalResults11013FieldTheoryJacobi’sNeccesaryConditionandSufficiency116iListofFigures1NeighborhoodSofX0..............................22NeighborhoodSofX0andaparticulardirectionH............

7、..23TwodimensionalneighborhoodofX0showingtangentatthatpoint.....54Theconstraintφ.................................65Thesurfaceofrevolutionforthesoapexample.................116Brachistochroneproblem.............................127AnarcconnectingX1andX2.....................

8、.....158Admissiblefunctionηvanishingatendpoints(bottom)andvariousadmissiblefunctions(top)..............