Euler as Physicist PDF
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Euler as Physicist Dieter Suisky Euler as Physicist 123 Dr.Dr.DieterSuisky Humboldt-Universita¨tzuBerlin Institutfu¨rPhysik Newtonstr.15 12489Berlin Germany [email protected] ISBN:978-3-540-74863-2 e-ISBN:978-3-540-74865-6 DOI10.1007/978-3-540-74865-6 LibraryofCongressControlNumber:2008936484 (cid:2)c Springer-VerlagBerlinHeidelberg2009 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 springer.com Preface In this book the exceptional role of Leonhard Euler in the history of science will be analyzed and emphasized, especially demonstrated for his fundamental contri- butions to physics. Although Euler is famous as the leading mathematician of the 18thcenturyhiscontributionstophysicsareasimportantandrichofnewmethods andsolutions.TherearemanybooksdevotedtoEulerasmathematician,butnotas physicist. In the past decade, special attention had been directed at the development of science in the 18th century. In three distinguished tercentenary celebrations in oc- casionofthebirthsofPierreLouisMoreaudeMaupertuis(1698–1759),Emiliedu Chaˆtelet(1706–1749)andLeonhardEuler(1707–1783)themeritsofthesescholars forthedevelopmentofthepost-Newtoniansciencehadbeenhighlyacknowledged. These events were not only most welcome to remember an essential period in the past,butarealsoanopportunitytoaskforthelonglastinginfluenceonthefurther developmentofscienceuntilpresentdays. Euler’s contributions to mechanics are rooted in his program published in two volumesentitledMechanicsorthescienceofmotionanalyticallydemonstratedvery early in 1736. The importance of Euler’s theory results from the simultaneous de- velopmentandapplicationofmathematicalandphysicalmethods.Itisofparticular interesttostudyhowEulermadeimmediateuseofhismathematicsformechanics and coordinated his progress in mathematics with his progress in physics. Euler’s mechanics isnotonlyamodelforaconsistentlyformulatedtheory,butallowsfor generalizationsofEuler’sprinciples. Thoughhispioneeringworkonmechanicshadanessentialinfluenceinthe18th century,itsimpactonthe19thcenturywasobscuredbytheoverwhelmingsuccess of his mathematical writings. Euler anticipated Mach’s later criticism of absolute motionandEinstein’sassumptionontheinvarianceoftheequationofmotioninin- ertialsystems.Itwillbedemonstratedthatevenproblemsincontemporaryphysics maybeadvantageouslyreconsideredandreformulatedintermsofEuler’searlyuni- fied approach. The interplay between physics and mathematics which appeared in the18thcenturywillbecomparedtothedevelopmentofphysicsinthe20thcentury, especiallytothedevelopmentofquantummechanicsbetween1900and1930. The reader of Euler’s works benefits from his unique ability to preserve math- ematical rigour in the analytical formulation of physical laws. The principles and v vi Preface methods found in the original sources may be advantageously compared to later developments, interpretations and reformulations. The extraordinary power of Eu- ler’sprogramandlegacyisduetothesuccessfulfrontiercrossingbetweendifferent disciplinespresentedinanexemplarilyclearterminology. IamgratefultothoseindividualswhodrawmyattentiontoEuler’soriginalwork anditscontemporaryinterpretation. I would like especially to thank Dr. Hartmut Hecht (Berlin-Brandenburgische AkademiederWissenschaften),Dr.PeterEnders(Berlin),Dr.PeterTuschik(Hum- boldtUniversity,Berlin),Dr.Ru¨digerThiele(UniversityofLeipzig),Prof.RobertE. Bradley(AdelphiUniversity,GardenCity),Prof.C.EdwardSandifer(WesternCon- necticut State University), Prof. Heinz Lu¨bbig (Berlin), Prof. Eberhard Knobloch (Technical University Berlin), Prof. Ruth Hagengruber (University of Paderborn), Dr.AndreaReichenberger(UniversityofPaderborn)andProf.BrigitteFalkenburg (UniversityofDortmund)formanystimulatingandencouragingdiscussions. In particular, I am indebted to Prof. Wolfgang Nolting (Humboldt University, Berlin)forstimulatingsupportoveraperiodofyears. Finally,IwouldliketothankmywifeIngridSuiskyforrenderingeveryhelpin preparingthemanuscript. Berlin DieterSuisky Introduction ReadEuler,readEuler,heisthemasterofusall. Laplace Eulerbeganhisextraordinaryscientificlifeinthethirddecadeofthe18thcen- tury and spent most of his career working in St. Petersburg and Berlin.1 His first paper was published in 1726, the last paper of his writings only in the mid of the 19thcentury.ForhisscientificcareerhewaswellpreparedandeducatedbyJohann Bernoulliwhointroducedhiminmathematicsbypersonalsupervision2since1720. His close relations to the Bernoulli family, especially to the older sons of Johann Bernoulli,lethimfinallysettleinSt.Petersburg3 in1727. Inthistime,thescientificcommunitywasinvolvedincontroversialdebatesabout basic concepts of mechanics like the nature of space and time and the measure of living forces, the priority in the invention of the calculus and the application of the calculus to mechanics. The promising Newton-Clarke-Leibniz debate between 1715 and 1716 was truncated by the death of Leibniz in 1716. The letters on the basic principles of natural philosophy exchanged by him with Clarke published in 1717 [LeibnizClarke] can be considered as a collection and listing of solved and 1 Eulerlivedfrom1707to1727inSwitzerland.InBasel,hestudiedmathematicsunderthesu- pervisionofJohannBernoulli(1667–1748).From1727to1740EulerworkedinSt.Petersburg. EulergrewuptogetherwithDanielBernoulli(1700–1782),NikolausIIBernoulli(1695–1726)and JohannIIBernoulli(1710–1790)Bernoulli,thesonsofJohannBernoulli.ThechangeofEulerto St.PetersburgwasmainlystimulatedbytheleaveofDanielBernoulliwhowenttoSt.Petersburg asaprofessorofmathematicsin1725.AfterasuccessfulandfruitfulperiodinSt.PetersburgEuler spentthedecadesfrom1741to1766inBerlinbeforehewentbacktoSt.Petersburg(1766–1783). EssentialcontributionsofEulertomechanicswerepublishedduringBerlinperiodbetween1745 and1765. 2“A.1720wurdeichichbeyderUniversita¨tzudenLectionibuspublicispromovirt:woichbald Gelegenheitfand,demberu¨hmtenProfessoriJohanniBernoullibekanntzuwerden,welchersich ein besonderes Vergnu¨gen daraus machte, mir in den mathematischen Wissenschafften weiter fortzuhelfen.PrivatLectionenschlugermirzwar(...)ab:ergabmiraber(...)alleSonnabend NachmittageinenfreyenZutrittbeysich,undhattedieGu¨temirdiegesammleteSchwierigkeiten zuerla¨utern,(...).”[Fellmann(Row)]EulerdictatedtheautobiographytohissonJohannAlbrecht inthesecondPetersburgperiod. 3“UmdieselbigeZeitwurdedieneueAkademie(...)inSt.Petersburgerrichtet,wohindiebey- dena¨ltestenSo¨hnedesH.JohannisBernoulliberuffenwurden;daichdenneineunbeschreibliche BegierdebekammitdenselbenzugleichA.1725nachSt.Petersburgzureisen.”[Fellmann(Row)] vii viii Introduction unresolvedproblemsinmechanicsandphilosophy.Oneofthecrucialpointsisthe debate on the nature of time, space and motion. This listing will be completed by theproblemsoriginatingfromthecontroversyonthepriorityoftheinventionofthe calculuswhichhadbeeninitiatedbyBritishscientistsin1690’sandcontinueduntil 1711[Meli].Furthermore,thescientificcommunitywasinvolvedinthelong-lasting debateonthetruemeasureoflivingforcesand,asaconsequence,splitintodiffer- ent schools. The post-Newtonian scientists were confronted with a legacy rich of newmethodsandsolutionsofoldproblems,butalsorichofopenquestionsandhad tostrugglewithacollectionofmostcomplicatedproblemswhichwere,moreover, closely interrelated preventing from the very beginning a merely partial solution. Nevertheless,suchpartialsolutionswereconstructedbytheseveralschoolswhich had been appeared, established and settled preferentially in France, England and Germany.4 Thedebatewasdominatedbytheseschools,theCartesians,theNewto- niansandtheLeibnizians,respectively.Thecommonproblemwastofindawayout fromthemergingofscientificandnon-scientificproblems. All the scientists were involved in the controversies as Euler reported 30 year later[Euler,E343].5 Theleadingscientistdecideditisnecessarytoreconsiderthe legacy of their predecessors without prejudice, i.e. independently of the country whereitwasproduced(compare[Chaˆtelet,Institutions]).Voltairedidagreatwork toaccustompeopleinFrancetotheideasofNewton[Voltaire,E´le´mens].Wolffwas interestinginthathissystembecomespopularinFrance. The profound education Euler received from Johann Bernoulli is not the only long lasting influence on his thinking and life. There is also the personal and sci- entific relation to Daniel Bernoulli which was of great influence on the scientific biography of Euler. The concept of his program for mechanics published in the Mechanica may be related to the discussion on the status of the basic law of me- chanics which had been later called Principe ge´ne´ral et fondamental de toute la me´caniquebyEuler[EulerE177].EulerreferredintheMechanica[EulerE015/016, § 152] to the relations between the change of velocity and forces proposed by Daniel Bernoulli published in 1722 [Bernoulli Daniel].6 Euler derived a general 4 In Germany the scientific discussions were dominated by the Leibniz-Wolffian school. Wolff (1679–1754) reinterpreted the Leibnizian system of monadology in terms of least elements of bodies.EulercriticizedtheLehrgeba¨udevondenMonaden[EulerE081]distinguishingcarefully betweentheoriginalLeibniziantheoryandtheWolffianinterpretation.However,judgingonthe meritsofthepredecessorswehavetotakeintoaccountthatmostofthewritingsofLeibnizand Newtonwereunpublishedin18thcentury.Therefore,thefollowershadtohavetoreconstructand re-inventtheoriginalversionsfromthepartswhichwerepublished.Asitcanbedemonstratedfor Leibniz’scontributionstologicwhichhadbeenonlyrediscoveredbyCouturatandRussellin19th century,thisisaverycomplicatedprocedure. 5 Euler commented on the debate on the nature of light [Euler E343, Lettre XVVII–XXI], the gravitation[EulerE343,LettreXLV,LVI–LXVIII],thesystemofmonadsandtheoriginofforces [EulerE343,LettreLXIX–LXXIX],Leibniz’smetaphysicsandotherphilosophicalsystems[Euler E344LettreLXXX–XCIX,CXXII–CXXXII]. 6 “152. Apparet igiturnon solumverumessehoctheorema,sedetiamnecessarioverum,itaut contradictioneminvolveretponeredc=p2dt vel p3dt aliamvefunctionemloco p.Quaeomnes cum Clar. Dan. Bernoullio in Comment. Tom. I. aeque probabiles videantur, de rigidis harum propositionumdemonstrationibusmaximeeramsollicitus.”[EulerE015/016,§152] Introduction ix relationbetweenthechangeofmotion,i.e.thechangeofvelocity,andtheforcesim- presseduponthebody.Moreover,Eulerclaimedthatthisrelationisnotonly“valid” (non solum verum), but also “necessarily valid” (sed etiam necessario verum). In 1743,treatingthesamerelationbetweentheaccelerationandforcesasadefinition, d’AlembertreferredtoDanielBernoulliandEulerandclaimedthatheisnotwill- inglytodecidethequestionwhetherthisprincipleisonlyjustifiedbyexperienceor anecessarytruth[d’Alembert,Traite´,§19]. Thenecessityofamechanicallawisconfirmedbyacomparisontomathematics, especiallytogeometry[EulerE181,§§1and2].Eulerpreservedthismethodolog- ical principle and repeated this approach to mechanics from 1734 also almost 30 yearslaterintheLettresa` uneprincessed’Allemagne[EulerE344,LettreLXXI]7 and in the Theoria motus corporum solidorum seu rigidorum [EulerE289]. The distinctionbetweennecessarystatementsandcontingentstatementshadbeenelab- orated by Leibniz who based his program for mechanics on the assumption that geometryhastobecompletedbyprincipleswhichexplaintheactionandthesuffer- ingofbodies[Leibniz,Specimen,I(11)].Thegeometricaltruthsareconsideredas necessarytruths,incontrasttothelawsinmechanicsorothersciencesrelatedtoex- periencewhichhadbeencharacterizedascontingenttruths[Leibniz,Monadology, §§ 31–36]. As a direct consequence it follows that, although mechanics is related to experience the theory has to be based on principles of the same reliability (or necessity)asmathematics.8 Theprogramformechanicshastobesupportedbythe transfer of mathematical principles to mechanics. Then, the mathematical princi- ples are not softened or violated,9 but form a constitutive part of the theory. As a consequence, mathematical and mechanical principles had been not only applied or transferred into the other discipline, but preferentially confirmed and mutually tested in their reliability and applicability. Newton demonstrated this program for thetransferofgeometricalprinciplestomechanics. After the invention of the calculus by Newton and Leibniz, the same problem arises for the transfer of the arithmetical principles of the calculus to mechanics. However,evenfortheinventorsofthecalculus,thisstepcanbenomeanstakenfor grantedsinceitwasalmostautomaticallynecessarytoaccepttheearlierCartesian programandbasicprinciplesofCartesianmethodology.Newtonstated: Menofrecenttimes,eagertoaddtothediscoveriesoftheAncients,haveunitedthearith- metic of variable with geometry. Benefiting from that, progress has been broad and far- reachingifyoureyeisontheprofusenessofoutput,buttheadvanceislessofablessing ifyoulookatthecomplexityoftheconclusions.Forthesecomputations,progressingby means of arithmetical operations alone, very often express in an intolerably roundabout 7“Quelquefonde´equisoitcetteloi,quipourroitallerdepairaveclesve´rite´sge´ome´triques,(...).” [EulerE343,LettreLXXI] 8 The mathematical methods are to be developed in conformity with the rigor known from the legacyoftheAncients(compareChaps.2,3,4and5). 9ThisprogramhasbeeninventedanddemonstratedbyNewton.Itcanreadofffromthetitleofhis treatisePhilosophiaenaturalisprincipiamathematica[Newton,Principia].Itisveryelucidating totakenoticefromEuler’sprogrampresentedinthetitlesMechanicasivemotusscientiaanalytice exposita[EulerE015/016]andTheoriamotuscorporumsolidorumseurigidorum[EulerE289]. x Introduction wayquantitieswhichingeometryweredesignatedbythedrawingofasingleline.[Mathe- maticalPapersofIsaacNewton4:421] EulerrenewedtheCartesianprogram.In1727,Eulercomposedananalyticalfoun- dationofthecalculus[Euler1727]whosebasicprincipleshadbeenpreservedinthe followingdecades[EulerE212].StimulatedandeducatedbyhissupervisorJohann Bernoulli,Eulercouldmadeimmediateuseofthetransmittedresultsoftheexam- ination and application of the new method by Leibniz and Johann Bernoulli since 1684.10 Analyzing Euler’s philosophical statements people claimed that Euler re- newedtheCartesiandualismbetweenbodyandsoul.However,comparingEuler’s basic assumptions on the nature of bodies with the original Cartesian version, a remarkable and strong difference can be observed. Euler introduced an important changeofDescartesconceptofbodies.Descartesclaimedthattheextensionisbasic propertiesofallbodies,statingresextensasivecorpus.Eulerintroducedaprogram formechanicsbasedontheconceptofbodiesofinfinitesimalmagnitude. Thoselawsofmotionwhichabodyobserveswhenlefttoitselfincontinuingrestormotion pertain properly to infinitely small bodies, which can be considered as points. (...) The diversityofbodiesthereforewillsupplytheprimarydivisionofourwork.Firstindeedwe shallconsiderinfinitelysmallbodies(...).Thenweshallattackbodiesoffinitemagnitude whicharerigid.[EulerE015/016,§98] ReadingEuler’sprogramandconsideringtheactualbackgroundformedbythestate ofaffairsinmechanicsandmathematicsinthefirsthalfofthe18thcentury,itfol- lows that Euler’s program for mechanics should be necessarily based on an addi- tionalprogram,aprogramforreinterpretation,applicationanddevelopmentofthe calculus invented by Newton and Leibniz. In 1734, this program was widely hid- denandonlyexplicitlyformulatedinthetreatiseInstitutionescalculidifferentialis [EulerE212]writtenin1748andpresentedonlyin1755.Thisrelationcanbecon- firmed by the reference to a statement in the Preface of Mechanica. Here, Euler stated that the shortcoming of the geometrical method is the lack of an algorithm which can be used for the modelling and calculation of problems which deviated onlyslightlyinsomedetailfromthestandardformulationandsolution.11 10 Similarly,inthe20thcentury,HeisenbergcouldmadeuseofthediscussionsbetweenPlanck, Bohr, Einstein and Sommerfeld on the foundation of quantum mechanics (compare Chap. 8). The supervisors had established the new discipline and prepared the problem for solution, but thedecisivestepbeyondthecommonlyacceptedclassicalframewasdonebytheyoungHeisen- berg[Heisenberg1925].SimilarlytoEulerwhodecidedtocomposemechanicswithoutgeometry, but solely analytically demonstrated, Heisenberg decided to chose a “mechanics without posi- tionsandpaths”[Heisenberg1925]makingonlyuseofthemethodsof“transcendentalalgebra” [Schro¨dinger,SecondAnnouncement].Heisenbergrejectedthe“positionsandpaths”,Paulidis- cardedthe“causality”. 11 Moreover,Eulerinventedalgorithmalsofortheeverydayapplicationsofmathematics.“Man pflegt na¨mlich mit der eigentlichen Arithmetik noch einige Regeln, welche in der allgemeinen Analysi oder Algebra ihren Grund haben, zu vereinigen, damit ein Mensch, welcher dieselbe erlernet,auchimStandesei,diemeistenAufgaben,soindemgemeinenLebenvorzufallenpflegen, aufzulo¨sen,ohneinderAlgebrageu¨btzusein.”[EulerE017Chap.1,§1]Thisbookwaswritten inparallelwiththeMechanicain1735andpublishedin1738.Obviously,themainprincipleisto inventedalgebraicmethodsbothforthesolutionsofproblemsinscienceandeverydaylife. Introduction xi That which is valid for all the writings which are composed without the application of analysisisespeciallytrueforthetreatisesonmechanics(∗).Thereadermaybeconvincedof thetruthsofthepresentedtheorems,buthedidnotattainasufficientclarityandknowledge ofthem.Thisbecomesobviousifthesuppositionsmadebytheauthorsareonlyslightly modified.Then,thereaderwillhardlybeabletosolvetheproblemsbyhisowneffortsif hedidnottakerecoursetotheanalysisdevelopingthesametheoremusingtheanalytical method.[EulerE015/016,Preface](∗)Newton’sPrincipia(1687) ThecompletetitleofEuler’sbookonmechanicsissimultaneousanabbreviated,but precise program for the application of the calculus in mechanics: Mechanica sive motus scientia analytice exposita, i.e. mechanics or the science of motion demon- stratedbymeansofanalyticalmethodsortheapplicationofthecalculus.12Here,the analyticalapproachisopposedtothegeometricaltreatment13 preferredbyNewton inthePrincipia.FollowingDescartesandNewton,Eulerconstructedmechanicson thebasicconceptofrestandmotionand,methodologicallyonthebasicdistinction betweeninternalandexternalprinciples. Euler’sprogramformechanicsissimultaneouslyalsoaprogramformathemat- ics underlying the analytical approach. This program had been developed in the treatise entitled Calculus differentialis written in 1727, but only published form the first time in 1983 [Euler1727], [Euler, Juschkevich]. By this approach, Eu- ler continued a tradition which can be traced back to his predecessors Descartes, Newton and Leibniz14 to develop simultaneously mathematics and mechanics with a preference to the mathematically established algorithms. However, there is an essential difference. Although Newton invented an arithmetical algorithm [Newton,MethodofFluxions],[Newton,Principia],hepreferredgeometricalmeth- ods for the confirmation of the analytically obtained results and specified, e.g. the independentvariabletoberelatedtoa“continuousflux”or“time”.Analternative foundation based on arithmetic had been discussed by Leibniz [Leibniz,Historia] whobasedthemethodonthecorrelatedoperationsformedbydifferencesandsums [Leibniz,Elementa],15 wherethefirstoperationresultsinadifferentiationwhereas thesecondoneresultsintheinverseoperation,calledintegration.16 Fromthevery 12Theprogramisrelatedtothefoundationofthecalculusintermsofalgebraicoperationswhich hadbeenlaterdevelopedbyEuler[EulerE212].ThefatherofsuchtypeoffoundationisLeibniz whoinventedtheoperationsofthecalculusasoperationrulesappliedtoquantities[Leibniz,Nova Methodus]. 13 The Eulerian program had been later continued and developed by Lagrange. The titleof his basicbookonmechanicsisMe´caniqueanalytique[Lagrange,Me´canique].Lagrangestressedthat hedidnotmakeuseofanyfigures. 14 ItseemstobereasonabletoincludefurtherArchimedes,GalileoandHuygens.Eulerreferred fortheexplanationoftheconservationofstatetoArchimedes’considerationontheequilibrium betweenbodies[Euler,E015/016§56]. 15 Leibniz considered both the operation from the very beginning, he invented also the formal rulesandthesignsforthedifferentoperationandcalledtheprocedurecalculusofdifferencesand summation,althoughthenameintegrationwasfinallydueJohannBernoulli. 16 ComparetheappendixtotheVarignon’slettertoLeibnizfromMay23,1702,entitledJusti- ficationduCalculdesinfinitesimalesparceluydel’Algebreordinaire[Leibniz,Mathematische Schriften,vol.4,pp.99–106].
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