Euler as Physicist

356 Pages·2009·3.695 MB·English

by  Dieter Suisky (auth.), Dieter Suisky (eds.)

Euler as Physicist
Dieter Suisky
Euler as Physicist
123
Dr.Dr.DieterSuisky
Humboldt-Universita¨tzuBerlin
Institutfu¨rPhysik
Newtonstr.15
12489Berlin
Germany
[e-mail]
ISBN:978-3-540-74863-2 e-ISBN:978-3-540-74865-6
DOI10.1007/978-3-540-74865-6
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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].