selected chapters in the calculus of variations - moser j

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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eldsandglobalminimals4

3、12.1Globalextremalelds.........................412.2Anexistencetheorem.........................442.3Propertiesofglobalminimals.....................512.4Aprioriestimatesandacompactnesspropertyforminimals....592.5Mforirrational,Mathersets...................67

4、2.6Mforrational...........................852.7ExercicestochapterII.........................923DiscreteSystems,Applications953.1Monotonetwistmaps.........................953.2Adiscretevariationalproblem.....................1093.3Threeexamples................

5、.............1143.3.1TheStandardmap.......................1143.3.2Birkhobilliard.........................1173.3.3DualBillard...........................1193.4Asecondvariationalproblem.....................12223.5MinimalgeodesicsonT........................1233

6、.6Hedlund'smetriconT3........................1273.7ExercicestochapterIII........................1343.8Remarksontheliterature.......................13734CONTENTS0.1IntroductionTheselecturenotesdescribeanewdevelopmentinthecalculusofvariationscalledAubry-Mather

7、-Theory.ThestartingpointforthetheoreticalphysicistAubrywasthedescriptionofthemotionofelectronsinatwo-dimensionalcrystalintermsofasimplemodel.Todoso,Aubryinvestigatedadiscretevariationalproblemandthecorrespondingminimals.Ontheotherhand,Matherstartedfromaspeci

8、cclassofarea-preservingannulusmappings,thesocalledmonotonetwistmaps.ThesemapsappearinmechanicsasPoincaremaps.SuchmapswerestudiedbyBirkhoduringthe1920'sinseveralbasicpapers.Mathersucceededin19