Table Of ContentSPRINGER BRIEFS IN PHYSICS
Costas J. Papachristou
Aspects
of Integrability
of Differential
Systems and Fields
A Mathematical
Primer for Physicists
123
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Costas J. Papachristou
Aspects of Integrability
of Differential Systems
and Fields
A Mathematical Primer for Physicists
123
Costas J.Papachristou
Department ofPhysical Sciences
Hellenic Naval Academy
Piraeus, Greece
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefs inPhysics
ISBN978-3-030-35001-7 ISBN978-3-030-35002-4 (eBook)
https://doi.org/10.1007/978-3-030-35002-4
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Preface
Thismonograph,writtenatanintermediatelevelforeducationalpurposes,servesas
anintroductiontotheconceptofintegrabilityasitappliestosystemsofdifferential
equations (both ordinary and partial) as well as to vector-valued fields. We stress
fromtheoutsetthatthisisnotatreatiseonthetheoryorthemethodsofsolutionof
differential equations! Instead, we have chosen to focus on specific aspects of
integrability that are often encountered in a variety of problems in Applied
Mathematics, Physics, and Engineering.
With regard to Physics, in particular, integrability is a subject of major impor-
tance given that most physical principles are expressed mathematically as systems
of differential equations. In Classical Mechanics, certain mathematical techniques
are employed in order to integrate the equations of a Newtonian or a Hamiltonian
system.Thesemethodsinvolveconceptssuchasconservationlaws,whichfurnisha
numberofconstantsofthemotionforthesystem.InElectrodynamics,ontheother
hand,theintegrability(self-consistency)oftheMaxwellsystemofequationsisseen
tobeintimatelyrelatedtothewavelikebehavioroftheelectromagneticfield.Inthe
staticcase,theintegrability(inthesenseofpath-independence)oftheelectricfield
leadstotheconceptoftheelectrostaticpotential.Finally,anumberofmethodshave
been developed for finding solutions of nonlinear partial differential equations that
areofinterestinMathematicalPhysics.Beforeembarkingonthestudyofadvanced
Physics problems, therefore, the student will benefit by being exposed to some
fundamentalideasregardingthemathematicalconceptofintegrabilityinitsvarious
forms.
The following cases of integrability are examined in this book: (a) path-
independenceoflineintegralsofvectorfieldsontheplaneandinspace;(b)integra-
tion of a system of ordinary differential equations (ODEs) by using first integrals;
and (c) integrable systems of partial differential equations (PDEs). Special topics
includetheintegrationofanalyticfunctionsandsomeelementsfromthegeometric
theoryofdifferentialsystems.Certainmoreadvancedsubjects,suchasLaxpairsand
Bäcklund transformations, are also discussed. The presentation sacrifices mathe-
maticalrigorinfavorofsimplicity,asdictatedbypedagogicallogic.
v
vi Preface
A vector field is said to be integrable in a region of space if its line integral is
independentofthepathconnectinganytwopointsinthisregion.Aswillbeseenin
Chap. 1, this type of integrability is related to the integrability of an associated
system of PDEs. Similar remarks apply to the case of analytic functions on the
complexplane,examinedinChap.2.Inthiscase,theintegrablesystemofPDEsis
represented by the familiar Cauchy-Riemann relations.
In Chap. 3, we introduce the concept of first integrals of ODEs and we
demonstrate how these quantities can be used to integrate those equations. As a
characteristic example, the principle of conservation of mechanical energy is used
tointegratetheODEexpressingNewton’ssecondlawofmotioninonedimension.
ThisdiscussionisgeneralizedinChap.4forsystemsoffirst-orderODEs,where
thesolutiontotheproblemisagainsoughtbyusingfirstintegrals.Themethodfinds
animportantapplicationinfirst-orderPDEs,thesolutionprocessofwhichisbriefly
described. Finally, we study the case of a linear system of ODEs, the solution of
which reduces to an eigenvalue problem.
Chapter 5 examines systems of ODEs from the geometric point of view.
Concepts of Differential Geometry such as the integral and phase curves of a
differential system, the differential-operator representation of vector fields, the Lie
derivative,etc.,areintroducedatafundamentallevel.Thegeometricsignificanceof
first-order PDEs is also studied, revealing a close connection of these equations
with systems of ODEs and vector fields.
Two notions of importance in the theory of integrable nonlinear PDEs are
Bäcklund transformations and Lax pairs. In both cases, a PDE is expressed as an
integrabilityconditionforsolutionofanassociatedsystemofPDEs.Theseideasare
briefly discussed in Chap. 6. A familiar system of PDEs in four dimensions,
namely,theMaxwellequationsfortheelectromagneticfield,isshowntoconstitute
a Bäcklund transformation connecting solutions of the wave equations satisfied by
theelectricandthemagneticfields.ThesolutionoftheMaxwellsystemforthecase
of a monochromatic plane electromagnetic wave is derived in detail. Finally, the
useofBäcklundtransformationsasrecursionoperatorsforproducingsymmetriesof
PDEs is described.
I would like to thank my colleague and friend Aristidis N. Magoulas for an
excellent job in drawing a number of figures, as well as for several fruitful dis-
cussions on the issue of integrability in Electromagnetism.
Piraeus, Greece Costas J. Papachristou
September 2019
Contents
1 Integrability on the Plane and in Space . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Simply and Multiply Connected Domains . . . . . . . . . . . . . . . . . . 1
1.2 Exact Differentials and Integrability. . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Line Integrals and Path Independence . . . . . . . . . . . . . . . . . . . . . 6
1.4 Potential Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Conservative Force Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Integrability on the Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Integrals of Complex Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Some Basic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Antiderivative and Indefinite Integral of an Analytic Function. . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 The Concept of the First Integral. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Integrating Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Higher-Order Differential Equations. . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Application: Newton’s Second Law in One Dimension. . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Systems of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . 37
4.1 Solution by Seeking First Integrals . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Application to First-Order Partial Differential Equations . . . . . . . . 43
4.3 System of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
vii
viii Contents
5 Differential Systems: Geometric Viewpoint. . . . . . . . . . . . . . . . . . . . 53
5.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Geometric Significance of the First Integral . . . . . . . . . . . . . . . . . 57
5.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4 Differential Operators and Lie Derivative. . . . . . . . . . . . . . . . . . . 61
5.5 Exponential Solution of an Autonomous System . . . . . . . . . . . . . 63
5.6 Vector Fields as Generators of Transformations . . . . . . . . . . . . . . 65
5.7 Geometric Significance of First-Order PDEs. . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6 Integrable Systems of Partial Differential Equations. . . . . . . . . . . . . 71
6.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Bäcklund Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 Lax Pair for a Nonlinear PDE. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.4 The Maxwell Equations as a Bäcklund Transformation. . . . . . . . . 76
6.5 Bäcklund Transformations as Recursion Operators . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Appendix A: Conservative and Irrotational Fields. .... .... ..... .... 85
Appendix B: Matrix Differential Relations ... .... .... .... ..... .... 89
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 93
Chapter 1
Integrability on the Plane and in Space
1.1 SimplyandMultiplyConnectedDomains
WebeginwithafewbasicconceptsfromTopologythatwillbeneededinthesequel.
AdomainDontheplaneissaidtobesimplyconnectedif,foreveryclosedcurve
C withinthisdomain,everypointoftheplaneintheinteriorofC isalsoapointof
D.Alternatively,thedomainDissimplyconnectedifeveryclosedcurveinDcanbe
shrunktoapointwithouteverleavingthisdomain.Ifthisconditionisnotfulfilled,
thedomainiscalledmultiplyconnected.
InFig.1.1,theregion(α)issimplyconnected,theregion(β)isdoublyconnected
while the region (γ) is triply connected. Notice that there are two kinds of closed
curvesinregion(β):thosethatdonotencirclethe“hole”andthosethatencircleit.
(Wenotethattheholecouldevenconsistofasinglepointsubtractedfromtheplane.)
Byasimilarreasoning,thetripleconnectednessofregion(γ)isduetothefactthat
therearethreekindsofclosedcurvesinthisregion:thosethatdonotencircleany
hole,thosethatencircleonlyonehole(nomatterwhichone!)andthosethatencircle
twoholes.
AdomainΩ inspaceissimplyconnected if,foreveryclosedcurveC insideΩ,
thereisalwaysanopensurfaceboundedbyC andlocatedentirelywithinΩ.This
(α) (β) (γ)
Fig.1.1 Threedomainsontheplane,havingdifferenttypesofconnectedness
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