Change and Variations: A History of Differential Equations to 1900

421 Pages·2021·5.382 MB·English

by  Jeremy Gray

Springer Undergraduate Mathematics Series
Jeremy Gray
Change and
Variations
A History of Differential Equations
to 1900
Springer Undergraduate Mathematics Series
AdvisoryEditors
MarkA.J.Chaplain,St.Andrews,UK
AngusMacintyre,Edinburgh,UK
SimonScott,London,UK
NicoleSnashall,Leicester,UK
EndreSüli,Oxford,UK
MichaelR.Tehranchi,Cambridge,UK
JohnF.Toland,Bath,UK
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Jeremy Gray
Change and Variations
A History of Differential Equations to 1900
JeremyGray
SchoolofMathematicsandStatistics
OpenUniversity
MiltonKeynes,UK
ISSN1615-2085 ISSN2197-4144 (electronic)
SpringerUndergraduateMathematicsSeries
ISBN978-3-030-70574-9 ISBN978-3-030-70575-6 (eBook)
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Preface
TheShapeoftheBook
Thisisabookonthehistoryofmathematics;itsbasicdynamicishistoricalandthere-
fore,uptoapoint,chronological.Itfollowstheprogressofanumberofideasthat
grew,sometimescametogether,andoftendevelopedrichandfascinatingbranches
and applications. At its core is an account of how the calculus of Newton and
Leibniz—thecalculusoffunctionsofasinglevariable—ledtoattemptstodevelopa
calculusoffunctionsofseveralvariableandhowthesenewmathematicalmethods
contributedtothestudy,firstofordinary,andthenofpartialdifferentialequations.
Ineachcase,therationaleforthatworkwaschieflytodevelopgeneralmethodsthat
couldtackleproblemsingeometryandmechanics(themotionsofsolidsandliquids
undertheactionofforces).
Thephysicalworldbeingacomplicatedplace,mostoftheapplicationsinvolved
partialdifferentialequations,andherethestorysoonalsobecamecomplicated.The
first-orderpartialdifferentialequationintwoindependentvariableswasinitiallydiffi-
culttosolve,andthisposedproblemsforthestudyofmorethantwoindependent
variables and for equations of higher order. Important work on the first-order case
wasdonebyLagrangeandMongebeforeCauchywasfinallyabletoshowthatsuch
equationsalmostalwayshaveasolution.Butthesecond-ordercasealmostimmedi-
atelyconfineditselftothreespecialcases,somewhatasEulerhadsuggested,andall
ofthem,aswewouldsay,linear.Thefirst,andsimplest,isthewaveequation(the
prototypehyperbolicequation),successfullytackledbyd’Alembert.Eulerregarded
theonelaterknownastheellipticcase(thekeyexamplebeingtheLaplaceequation)
asbeingbeyondcurrentmethods.Finally,thecasewecallparabolicfellthrougha
gapinhisapproach,andstrangelylittlewassaidaboutitbeforeFourierdealtwith
thecanonicalexample:theheatequation.Atthispoint,asignificantdeparturefrom
thetheoryofordinarydifferentialequationsopenedup:theneedtopayattentionto
initialorboundaryconditions.However,thisissuewastoremainobscureforseveral
decades.
v
vi Preface
Euler quickly showed that linear ordinary differential equations with constant
coefficientscanbesolvedsystematically.Othertypesofordinarydifferentialequa-
tionswerestudiedintheeighteenthcentury,butthestoryispiecemeal,andinstead,
Ichosetogivejustoneexampleofthehistoryofordinarydifferentialequations:the
hypergeometric equation from Gauss to Riemann, Schwarz, and Poincaré. This is
one of the glories of the subject, bringing together early ideas about group theory,
complexfunctiontheory,andthethen-novelhyperbolicornon-Euclideangeometry.
Sohowisallthismaterialorganisedinthisbook?Chapter1connectsthecalculus
toproblemsinordinarydifferentialequationsandismirroredbyChaps.3and5in
whichthecalculusofseveralvariablesisdevelopedandthefirstpartialdifferential
equationsarestudied.Thenitisafairlystraightrunthroughtopicsinpartialdifferen-
tialequationtheoryinChaps.6,8,10,13,17–20.Thisallowsustoseehowthework
ofEuler,d’Alembert,andafewothersrewroteNewton’sPrincipiaMathematicafor
theeighteenth–twentiethcenturies.Thestoryofthehypergeometricequationoccu-
piesChaps.11and14–16becauseitmuststartin1812withGaussandbecauseonce
itgetsgoingitseemsridiculoustobreakitup.WhatinterveneshereisChap.12on
Cauchy’sdemonstrationoftheexistenceofsolutionstoordinarydifferentialequa-
tions and Chap. 13 on Riemann’s geometric version of complex function theory,
whichisneededforthesubsequentthreechapters.
What of the chapters not yet referred to? Chapter 2 describes the start of the
calculus of variations, and Chap. 7 takes that subject further into the eighteenth
century.Chapter4documentsothersuccessesofthepartialdifferentialcalculusin
studyingnaturalphenomenaotherthanthewaveequation.(Thereisalsoasurprising
linktothehypergeometricequation.)Chapters9and21areopportunitiesforrevision;
whenIgavethecourseIusedtheselecturestodiscusstheassessmentonthecourse
sofar.
Theremainingchaptersmoveintowhatmaybelessfamiliarmaterial.Riemann’s
studyof shock waves; Riemann and Weierstrasson minimalsurfaces;the workof
ThomsonandStokesonthetelegraphist’sequationandthelayingofthetrans-Atlantic
cable;alookatthefirstninieteenth-centuryattemptstorigorisethecalculusofvari-
ations;theeventualintroductionofthefundamentaltrichotomy(elliptic,parabolic,
hyperbolic)forsecond-orderlinearpartialdifferentialequationsandthefirstgeneral
existencetheoremsintheellipticandhyperboliccasesincludingHadamard’sinsis-
tenceofthedistinctionbetweeninitialandboundaryvalueproblems.Twochapters
look at how Jacobi used Hamilton’s ideas to create Hamilton-Jacobi theory and
subsequentattemptstogeometrisemechanics,andtheconnectiontothesolutionof
first-orderpartialdifferentialequations.
Allthismaterialhasacertaincoherencethatisworthspellingout.Ordinarydiffer-
entialequationsgrewoutof,oralongside,problemsinevaluatingintegrals,whichis
whywestilltalk,confusingly,ofintegratingadifferentialequationanditssolutions
as its integrals. It was soon recognised that the solution to an ordinary differential
equation was a family of functions and an individual solution could be specified
bymeansofsomeinitialconditions.So,itwasnaturalwhendifferentialequations
with several independent variables were investigated that the earliest researchers
(JeanleRondd’Alembert,LeonhardEuler,PierreSimonLaplace,andJoseph-Louis
Preface vii
Lagrange)thoughtofthesepartialdifferentialequationsinthesameway,andlooked
fortechniquesthatwouldproduceaformulaforthegeneralsolution(however,they
seldom also discussed an auxiliary process of fitting the general solution to some
initialconditions).Partofthestoryhereisthegradualrecognitionthatthisisnotthe
rightwaytothinkofpartialdifferentialequations.Rather,itisadialoguebetween
thegeneralmethodsandtheinitialorboundaryconditionsthatiscentral,andwhich
underpinsthecrucialdistinctionbetweentheellipticandhyperbolictypestowhich
formalmethodsareblind.Asweshallsee,thisexplainstheproblematicwayinwhich
complexvariableswerefirstused.
Itisalsointerestingtoseehowquestionsofrigourenterthestoryinawaythat
is immediately important and does not appear as the whim of an analytic pedant.
The ad hoc methods that can be used to solve a partial differential equation (such
astheseparationofvariables)naturallyraisethequestionoftheuniquenessofthe
solutions that is important in applications. The need for power series to converge
forcesaheavyrelianceon(realorcomplex)analyticmethodsthathas,ultimately,to
beoutflanked.
AdvicetoStudents
Thereareseveralimportantthingsbeingdescribedinthiscourse,anditmayhelpto
rememberwhattheyarebeforeimmersingyourselfinthetechnicaldetails.Newton’s
workisacaseinpoint.Eventhoughweshallonlyskimitssurface,itisclearthat
this is a remarkable achievement, one that it took more than a century to confirm,
and some of the best work of the twentieth century to surpass. Newton’s account
of the motion of the Moon and the planets does not rest on the calculus, still less
differentialequations,buteveryoneafterhimturnedtothecalculus,andEulergave
everyonethemeanstowritecelestialmechanicsthatwayeverafter.Mathematical
accountsoffundamentalphysicalprocesses—gravity,theproductionandspreadof
sound,thepropagationofheatandofelectricsignals—areamongthesuccessesof
thetheoryofpartialdifferentialequations.
Butforthecalculustodothiswork,mathematicianshavetohavetheconfidence
thatitdoeswork.Thisispartlyamatterofrigour,andindeeditissatisfyingtosee
that so many of the questions that are dealt with in courses on pure analysis arose
incontextswhereapractical,oratleastaphysical,answerdependedonthequality
ofthereasoning.Lessobviously,butperhapsmoreinterestingly,itisworthseeing
whateventhebestmathematiciansdidwithdifficultproblems.Argumentshaveto
berigorous—ultimately.Beforethentheyhavetobesomeorallofconvincing,intel-
ligible,general,plausible,andapplicable.Likewise,solutionshavetobeanumber
ofthingsapart,ideally,frombeingright:amongthecriteriaare,ononeoccasionor
another,computable,accurateenough,intelligible,complete,andunique.Justasan
argumentmightgettotheheartofaproblemorsomehowmerelywork,asolution
canbetrulyinformativeormerelyaformula.Allofthisisondisplayhere.And,of
course,sometimesanequationwithnoanswerinsightcanstillseemtobeavaluable
advance.
viii Preface
In this sense, the most momentous event on display here in the first half of the
courseisthatthecalculus,intheformofdifferentialequations,bothordinaryand
partial,candeliversomuchinsight.Thecomparablechangeinthesecondhalfofthe
course,asIhintedabove,isthetransformationinwhatasolutionistakentobe.For
partialdifferentialequationsthisistherisetoequalimportancewiththeequationof
theboundaryorinitialconditions,coupledasitiswithaprofoundclassificationof
theseequationsintotypes.Theneedforrigourplayeditspartinthesedevelopments
when appeals to the so-called generality of analysis and its supposed algebraic or
formalbasisbegantofail.
Thisisnotasetoflecturesinwhichepsilonsanddeltas,nsandNsdanceevermore
intricately,butthisshouldnotsuggestthatwhenmathematicsisapplied—whatever
thatmightmean—standardsdrop.Mathematiciansweredoingtheirbestatalltimes
to get it right, although we can observe different ways in which they honoured
thatcommandment.Thedifficultmathematicsherecomesfromthedifficultyofthe
problems:apartialdifferentialequationisadifficultthingtounderstand,harderthan
anordinarydifferentialequation,andharderthanmanyanearlyinvestigatorrealised.
Qualitativeargumentsareoftenharderthanquantitativeones,iflesstechnical.
Thechallengeyoufaceistogetasenseofthatstruggle,ofthedifficulty,andhow
itwastackled.
Being a historian of mathematics means attending to mathematics on its own
termsaswellasours,andseeingitinthecontextofitstime.Whatwasknown,what
wasthoughttobetrue?Whenamathematiciantacklesaproblem,youask:Howhad
otherproblemslikethisonebeentackled,howweretheytackledafterthisone?Is
theanalysisoftheproblemconvincing,isthesolutioninformative?What,intheend,
werethesepeopletryingtodo?
HistoriographicalRemarks
There are several existing accounts of the history of calculus, and a number of
specialistbooksandarticlesonparticularaspectsofthathistory.Thecontributions
of Newton and Leibniz, Euler, Lagrange, Fourier, Cauchy, Riemann, Weierstrass,
Poincaré,andHadamardhavebeenstudiedinsomedepth;varioustopics,suchasthe
waveequation,theheatequation,Laplace’sequation,andtheDirichletproblemhave
beenlookedatinsomedetail,althoughnotalwaysaftertheoriginalbreakthroughs
were made. But there is no general history of differential equations, ordinary or
partial.Historiesofthecalculusdwellonthestoryoftherigorisationofthecalculus
and the creation of modern (or, rather, nineteenth century) mathematical analysis,
but tend to marginalise the story of what made the calculus valuable: the capacity
it gives mathematicians and scientists to formulate and solve problems across the
fieldsofphysicsandgeometry.Historianshavetendedtoforgetthatwhatmadethe
calculus worth all the efforts to understand it was not ideas about infinitesimals,
differentials,limits,andthelikethatwereintroducedtoexplainandjustifyit,butits
manysuccessesinprovidinganunderstandingofthenaturalworld,fromthemotion
Preface ix
oftheplanetstothetransmissionofelectricsignals,andinextendingthepowersof
geometry.
There have been a few notable departures from this scholarly regime in recent
years.CraigFraserandJesperLützenhavesteadilyenrichedourunderstanding,and
otherhistorians(TomArchibald,JuneBarrow-Green,UmbertoBottazzini,Christian
Gilain,andTomHawkins,amongthem)havedealtwithvariousaspectsofthedevel-
opment of the theory of differential equations as it entered into the larger pictures
theywereexploring.
WhatThisBookisNot
ThelargestomissionistheworkofMaxwellandtheequationsnamedafterhim,but
itseemedtomethatthemoderntheory,andthephysicalexperimentsthatitexplains,
are largely unknown to mathematics students and would have entailed too great a
detourtobringtolife.Inaddition,Maxwell’sideasaboutthephysicsinvolvedare
notthemodernones—andfamously,noContinentalphysicistclaimedtounderstand
them—soitwouldhavebeenimpossibletodothemjusticeinthespaceavailable.
DisappointedreadersshouldconsultBuchwald’sFromMaxwelltoMicrophysics.For
muchthesamereasons,IwasunabletodealwithhydrodynamicsandtheNavier–
Stokesequations,butreadersmayalwaysturntoDarrigol’sWorldsofFlow.
Another topic that is wholly missing is the use of perturbative methods. Most
differential equations, and systems of such equations, that arose in practice could
only be tackled by the method of undetermined coefficients. The idea was to start
from a simplified version of the problem at hand that, however, admitted an exact
solution, and to seek the solution to the solution to the actual problem by adding
more terms to cope with the increased complexity. These might take the form of
powerseries,orlatertrigonometricseries,whichwerefittedtowhatdatatherewas,
especially in the important subject of astronomy, and their coefficients adjusted to
refinepredictionsandexplainothereffects.
AnotherimportanttopicthatitwouldbegoodtohaveincludedisSturm-Liouville
theory,butthereisalreadyanexcellenthistoricalaccountinLützen’sJosephLiouville
(1809–1882):MasterofPureandAppliedMathematics,andIthoughtitbettertoadd
tothestockofhistoricalinformationaboutthedevelopmentofdifferentialequations.
IwouldverymuchhavelikedtohaveconcludedthecoursewithPoincaré’sideas
aboutflowsonsurfaces,andhisbrilliantextensionofthesetechniquesinhisfamous
memoironthethree-bodyproblem,whichwouldhavemadeanattractiveconnection
backtoNewton’sPrincipiaMathematica,buttheresimplywasnoroom.However,
thereareexistingaccountsofthissubject.1
And,Iadmit,thewishtotrytosaysomethingaboutthehistoryofpartialdiffer-
entialequations,surelythelargestomissioninthehistoryofmodernmathematics,
alsoplayedapartinmydecisionsaboutwhattoinclude.
1Seemostinformatively(Barrow-Green1997),andalso(Gray2013),and(Verhulst2012).