The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century PDF
Preview The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century
Sources and Studies in the History of Mathematics and Physical Sciences Managing Editor J.Z. Buchwald Associate Editors J.L. Berggren and J. Lu¨tzen Advisory Board C. Fraser, T. Sauer, A. Shapiro For other titles published in this series, go to http://www.springer.com/series/4142 Emmy Noether (1882–1935) (photograph courtesy of the Emmy Noether Foundation, Bar Ilan University) Yvette Kosmann-Schwarzbach The Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E. Schwarzbach 123 Yvette Kosmann-Schwarzbach Bertram E. Schwarzbach Centre de Mathématiques Laurent Schwartz (Translator) École Polytechnique 91128 Palaiseau France [email protected] ISBN978-0-387-87867-6 e-ISBN978-0-387-87868-3 DOI10.1007/978-0-387-87868-3 SpringerNewYorkDordrechtHeidelbergLondon MathematicsSubjectClassification(2010):01-02, 22-03, 22E70, 49-03, 49S05, 70-03, 70H03, 70H33, 83-03 ©SpringerScience+BusinessMedia,LLC2011 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnec- tionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orby similarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) InmemoryofYseult ′′ d r wholikedscienceaswellashistory Preface Whatfollowsthusdependsupon acombinationofthemethodsof theformalcalculusofvariations andofLie’stheoryofgroups. EmmyNoether,1918 Thisbookisaboutafundamentaltextcontainingtwotheoremsandtheirconverses whichestablishedtherelationbetweensymmetriesandconservationlawsforvaria- tionalproblems.Thesetheorems,whoseimportanceremainedobscurefordecades, eventually acquired a considerable influence on the development of modern theo- reticalphysics,andtheirhistoryisrelatedtonumerousquestionsinphysics,inme- chanicsandinmathematics.Thistextisthearticle“InvarianteVariationsprobleme” byEmmyNoether,whichwaspublishedin1918intheGo¨ttingerNachrichten,and ofwhichwepresentanEnglishtranslationinPartIofthisbook. ThetranslationofNoether’sarticleisfollowed,inPartII,byadetailedanalysis ofitsinception,aswellasanaccountofitsreceptioninthescientificcommunity.As thebackgroundtoNoether’sresearch,wesketchsomedevelopmentsinthetheory ofinvariantsinthenineteethcenturywhichculminatedinthedefinitionandstudy of differential invariants, we discuss several works in mechanics dating from the beginningofthetwentiethcenturyinwhichSophusLie’sinfinitesimalmethodsin thetheoryofgroupsbegantobeapplied,andweshowthattheimmediatemotiva- tionforherworkwasrelatedtoquestionsarisingfromEinstein’sgeneraltheoryof relativityof1915.WethensummarizethecontentsofNoether’sarticleinmodern language. In the subsequent chapters, we review the way in which Noether’s con- temporaries,themathematiciansFelixKlein,DavidHilbertandHermannWeyl,and thephysicistsEinsteinandWolfgangPauli,acknowledgedorfailedtoacknowledge hercontribution;thenweoutlinethequitedifferentdiffusionsofherfirstandsecond theorems. Finally, we outline the genuine generalizations of Noether’s results that begantoappearafter1970,inthefieldofthecalculusofvariationsandinthetheory ofintegrablesystems. ThepresenteditionisbasedonthesecondeditionofLesThe´ore`mesdeNoether. Invariance et lois de conservation au XXe sie`cle (Palaiseau: E´ditions de l’E´cole Polytechnique, 2006). For this English edition, the French text has been consider- ablyrevisedandaugmented,withmuchnewinformationandadditionalreferences. Paris,July2010 vii Acknowledgments Itisapleasuretothankthemanycolleaguesandfriendswhohavemadethewriting of this book possible. When the French version first appeared in 2004, it was at theinsistanceofPierreCartierwhosesubsequentsupportwasunfailing,withAlain Guichardet’sgeneroushelp,andwiththemuchappreciatedcontributionofLaurent Meersseman who had translated the “Invariante Variationsprobleme” into French andwrittenadraftofthefirstsectionofacommentary. During the preparation of the French and English texts, I received information andassistanceonvariousquestionsfromAlainAlbouy,SophieBade,HenriBesson, KatherineBrading,HarveyBrown,LeoCorry,ThibaultDamour,JeanEisenstaedt, BenjaminEnriquez,RobertGergondey,PatrickIgle´sias-Zemmour,MichelJanssen, Christoph Kopper, Franco Magri, Hartmann Ro¨mer, Peter Roquette, Volodya Rubtsov,DavidRowe,ErhardScholz,Jean-MarieSouriau,JimStasheff,Alexander Vinogradov, Salomon Wald, Stefan Waldmann and Scott Walter. To all of them— and to any colleagues whose names I may have omitted inadvertently—I am very grateful. I especially thank Peter Olver who made many useful remarks on the manuscriptofthefirstFrenchedition,basedonhisthoroughknowledgeofthesub- ject,andagainansweredmyquestionswhenIwaspreparingthesecondeditionin 2006. ThethreerefereesforthefirsteditionoftheFrenchtextmadevaluable rec- ommendationsandsuggestions,fromwhichIbenefitted.ThehelpIreceivedfrom David Schimmel in translating several difficult German texts remains a treasured memoryofthislearnedfriendwhodiedinaroadaccidentin2007. Thanks are also due to Fre´de´ric Zantonio at the Centre Polyme´dia de l’E´cole Polytechnique who helped me with the reproductions. The editors of Springer New York, Ann Kostant, Hans Koelsch, and Elizabeth Loew, extended their sup- portduringthepreparationofthisbook,forwhichIthankthemcordially. I am grateful to the institutions which have allowed me to reproduce docu- mentsintheircollections:theNiedersa¨chsischeStaats-undUniversita¨tsbibliothek, Go¨ttingen,theEinsteinarchiveinJerusalem,andthearchiveofBrynMawrCollege, andtoPrincetonUniversityPressforpermissiontoquotefromseveralvolumesof TheCollectedPapersofAlbertEinstein. Lastbutnotleast,Ithankthetranslator,BertramE.Schwarzbach, forhiswork andforhisadviceonhowtoclarifyandimprovetheFrenchtext. YvetteKosmann-Schwarzbach July2010 ix
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