Table Of ContentProgress in Mathematical Physics
Volume64
Editors-in-Chief
AnneBoutetdeMonvel,UniversitéParisVIIDenis Diderot, France
Gerald Kaiser, Center forSignalsand Waves,Austin,TX, USA
Editorial Board
C. Berenstein, UniversityofMaryland,CollegePark, USA
SirM.Berry, UniversityofBristol, UK
P. Blanchard, University of Bielefeld, Germany
M.Eastwood,UniversityofAdelaide, Australia
A.S. Fokas,University of Cambridge, UK
D. Sternheimer, Université de Bourgogne, Dijon, France
C. Tracy,UniversityofCalifornia,Davis, USA
Forfurthervolumes:
h ttp://www.springer.com/series/4813
Bruno Cordani
Geography of Order
and Chaos in Mechanics
Investigations of Quasi-Integrable
Systems with Analytical, Numerical,
and Graphical Tools
Bruno Cordani
Italy
[email protected]
Please note that additional material for this book can be downloaded from http://extras.springer.com
ISBN 978-0-8176-8369-6 ISBN 978-0-8176-8370-2 (eBook)
DOI 10.1007/978-0-8176-8370-2
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2012948333
Mathematics Subject Classification (2010): 70F05, 70F10, 70F15, 70H08, 70H33, 70K65
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Alla mia famiglia,
che mi ha supportato e sopportato
Preface
Knowledgewithouttheoryisblind
andwithoutpracticeisvoid.
—I.Kant
T
heNewtonianprogram,wellknownbyeverystudent,isconceptuallysim-
ple and attractive: given a mass distribution and the forces acting on it,
write the differential equations arising from the fundamental law of dy-
namicsandsolvetheminordertoobtainthemotion. Unfortunately,things
arenotsosimple,andinthecourseoftheprogramoneencountersatleast
twoessentialandunavoidableobstacles.
First,wearenotableingeneraltosolve(technically,tointegrate)asys-
tem of differential equations. Yes, every young student has learned how
to tackle the harmonic oscillator, the two-body problem, or the free rigid
body. But it is discouraging that these systems, along with a few others
discoveredmainlyinthesecondhalfofthepastcentury,exhaustthesmall
listofintegrablesystems.
Butevenifonepossessedmagicallyananalyticalformulagivingexactly
thetimeevolution,itwouldstillbescarcelyusefulforvariousreasons. For
example,themotionisingeneralverycomplicated,andfollowingthesolu-
tioninitswanderingdoesnotgivevaluableinformationaboutthenatureof
thephenomenon. Whatismore,apossibleregularityinthemotionisdiffi-
culttodetectbysimplyinspectingthedynamicalevolutionofthephysical
coordinates. Another frequent difficulty is the extreme sensitivity to the
initial conditions (“butterfly effect”), which in practice makes the concept
ofsolutionitselfmeaningless. Butthisshouldnotcomeasasurprise: after
vii
viii Preface
all,everybodyhasfeltasenseoffrustrationlookingatthenumericalsolu-
tionofsomethree-dimensionalsystems,beingunabletoextractameaning
fromtheentangledtrajectoryappearingonthemonitor.
The aim of this book is to show how to overcome these difficulties and
grasptheessenceofthedynamicsintheparticularbutveryimportantand
significantcaseofquasi-integrablesystems,i.e.,integrablesystemsslightly
perturbed by other forces. A paradigmatic case is the solar system, where
the perturbations are the interactions among the planets. Besides their
practical importance, these systems are also extremely interesting from a
mathematicalpointofview,exhibitinganintricateandfascinatingstructure
knownasthe“Arnoldweb.”
In the book these systems will be studied both from the analytical and
thenumericalpointofview. Withregardtothefirstpoint,Ithinkthatitis
impossible to overestimate the importance of the role played by the sym-
plecticstructureofthephasespaceor,inmoretraditionallanguage,bythe
Hamiltonianformoftheequationsofmotion. Thisstructureisthenatural
one of the phase space, exactly as the Euclidean structure is the funda-
mentaloneofourphysicalspace. Itisthesymplecticstructurethatforces
the solutionsofthe integrable systems to evolve linearlyon tori(products
of circles) with some fundamental frequencies, providing the framework
withoutwhichthetwomaintheoremsofperturbationtheory,i.e.,KAMand
Nekhoroshev, could not even be enunciated. It is thus not surprising that,
asalreadydevisedbythegreatfoundersoftheanalyticalmechanicsinthe
nineteenthcentury,oneshouldconstantlyutilizethesymplectic(canonical)
coordinatesadaptedtothefoliationintori,theaction-anglevariables,which
deeply reveal the hidden properties of the perturbed motions. Exploiting
the advanced techniques of perturbation theory, many examples of reduc-
tion tonormalformwillbegiven, i.e., toanintegrable, henceapproximate
formthathoweverreproducesthetruedynamicswell.
Inordertocomparetheapproximatewiththetruedynamicsoneneeds
numerical methods. In the book I present some tools recently introduced:
the Frequency Modified Fourier Transform (a refinement of the Discrete
Fourier Transform), the Wavelets (which allow one to find the instanta-
neous frequency) and the Frequency Modulation Indicator (which detects
the distribution of the resonances among the fundamental frequencies).
Thereadermayalsofindmanyfiguresthatwellillustratetheeffectiveness
ofthemethodsand,aboveall,therelativesoftware. Thisissurelythemain
feature of the book: the reader himself can and is encouraged to repro-
ducethevariousfiguresofthebookandexperimentwithothersituations,
exploring the details of various quasi-integrable systems. I am convinced
that the union of theory and practice is the main route to try to master an
argumentthatisconsidereddifficult.
Butamoreprofoundmotivationinresortingtonumericalcomputations
arisesfromsomelackofreliabilitythattoacertainextenteverymathemati-
Preface ix
cian experiences when facing a theorem proof that is particularly lengthy
and intricate, and that looks more like a rhetoric speech to persuade the
readerthanthegraniticstatementofanunquestionabletruth. Thenumeri-
calexperimentsbecomesoanessentialcompletionofthetraditionalproof,
reversingTruesdell’sthesisof“thecomputer: ruinofscienceandthreatto
mankind.”
I’d like to make it clear that no knowledge on computer programming
is needed in order to use the software: you only have to access directly to
a MATLAB installation or, subordinately, to install a free reader. Indeed,
the programs support a graphical user interface and require one only to
click on buttons and menu having a hopefully clear meaning: see the final
appendices to the book. The supplied programs in the accompanying CD
canbedownloadedasanisoimagefromthepublisher’swebsitebyentering
thebook’sISBN(978-0-8176-8369-6)intohttp://extras.springer.com/
andarethefollowing.
(i) POINCAREprogramanalyzessymplecticmapswiththeaidoftheFre-
quencyModulationIndicator.
(ii) HAMILTONprogramanalyzesHamiltoniansystemswiththeFrequency
ModulationIndicator.
(iii) LAGRANGE program regards the Lagrange points in the three-body
problem.
(iv) KEPLERprogramstudiestheperturbationsoftheKeplerproblem.
(v) LAPLACEprogramconcernsthedynamicsofasolarsystem.
I used part of the material presented here in some courses on Celestial
Mechanics, Hamiltonian Systems, and Perturbation Theory, addressed to
advanced undergraduate students. I think that the book may serve as an
introductiontospecialisticliteratureandtoaseriousstudyofperturbation
theory, with particular emphasis on the KAM and Nekhoroshev theorems.
The two theorems are proved in the book skipping some details, like the
technical proof of bounding inequalities, which in a first approach (and
alsoinasecond)aremoredistractingthanilluminating,andtryinginstead
to stress the conceptual points. But I hope that professional researchers
mayalsofindthisbookuseful,thankstoitsenclosedsoftware.
Briefly, the plan of the work is the following. In Chapter 1 a somewhat
detailed account of the whole book is given, which should also help the
reader to not lose the thread of the argument. Chapter 2 contains the ba-
sicconceptsofdifferentialgeometry,Liegroups,andanalyticalmechanics,
which Chapter 3 applies to perturbation theory. Chapter 4 deals with nu-
merical integration of ordinary differential equations and Chapter 5 with
some tools useful to numerically detect order and chaos. The final four
x Preface
chaptersaredevotedtotheapplications,i.e.,totheperturbationsoftheKe-
plerproblem,asthehydrogenatominanelectricandmagneticfield,andto
theplanetaryproblem. Theseconcreteapplicationsarenotonlyphysically
interesting but are also significant examples of how to investigate in gen-
eralquasi-integrableHamiltoniansystems,combiningthetechniquesofthe
reductiontonormalformwiththenumericalanalysisofhoworder,chaos,
andresonancesaredistributedinphasespace.
Itisalwaysusefultolistentoseveraldifferentvoicesonthesameargu-
ment. Threebooksinparticulararehighlyrecommendable: Celletti(2010),
Morbidelli(2002),andFerraz-Mello(2007). Moreorlesstheycoverthesame
topics of the present book, with a major emphasis on the applications but
withoutincludinganysoftware. AgoodintroductiontothisbookisTabor
(1989).
I thank very much the two anonymous referees for all the useful com-
mentsandsuggestions,whichI’ve(almost)fullyincluded.
Finally, I’d like to express my sincerest thanks to Tom Grasso and Ben
CroninatBirkhäuserfortheirhighlyprofessionalandefficienthandlingof
thisproject.
Milano,June2012 B.Cordani
