moser j. selected chapters of variational calculus (lecture notes, 2002)(149s)

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1、SelectedchaptersinthecalculusofvariationsJ.Moser2Contents0.1Introduction...............................40.2Ontheselecturenotes.........................51One-dimensionalvariationalproblems71.1Regularityoftheminimals.......................71.2Examples....................

2、............131.3TheacessoricVariationalproblem..................221.4Extremaleldsforn=1........................271.5TheHamiltonianformulation.....................321.6ExercicestoChapter1.........................372Extremaleldsandglobalminimals412.1Globalextremaleld

3、s.........................412.2Anexistencetheorem.........................442.3Propertiesofglobalminimals.....................512.4Aprioriestimatesandacompactnesspropertyforminimals....592.5Mforirrational,Mathersets...................672.6Mforrational..................

4、.........852.7ExercicestochapterII.........................923DiscreteSystems,Applications953.1Monotonetwistmaps.........................953.2Adiscretevariationalproblem.....................1093.3Threeexamples.............................1143.3.1TheStandardmap.........

5、..............1143.3.2Birkhobilliard.........................1173.3.3DualBillard...........................1193.4Asecondvariationalproblem.....................12223.5MinimalgeodesicsonT........................1233.6Hedlund'smetriconT3........................1273.7Exer

6、cicestochapterIII........................1343.8Remarksontheliterature.......................13734CONTENTS0.1IntroductionTheselecturenotesdescribeanewdevelopmentinthecalculusofvariationscalledAubry-Mather-Theory.ThestartingpointforthetheoreticalphysicistAubrywasthedescr

7、iptionofthemotionofelectronsinatwo-dimensionalcrystalintermsofasimplemodel.Todoso,Aubryinvestigatedadiscretevariationalproblemandthecorrespondingminimals.Ontheotherhand,Matherstartedfromaspecicclassofarea-preservingannulusmappings,thesocalledmonotonetwistmaps.Thesemap

8、sappearinmechanicsasPoincaremaps.SuchmapswerestudiedbyBirkhoduringthe1920'sinseveralbasicpapers.Mathersucceededin19