Hamiltonian Dynamical Systems and Applications

450 Pages·2008·5.67 MB·English

by  Walter Craig

Hamiltonian Dynamical Systems
and Applications
NATO Science for Peace and Security Series
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Series B:Physics and Biophysics
Hamiltonian Dynamical Systems
and Applications
edited by
Walter Craig
McMaster University, Hamilton, ON, Canada
Published in cooperation with NATO Public Diplomacy Division
Proceedings of the NATO Advanced Study Institute on
Hamiltonian Dynamical Systems and Applications
Montreal, Canada
18–29 June 2007
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Preface
Thisvolumeisacollectionoflecturenotesfromthecoursesthatweregivenduring
the 2007 Séminaire de Mathématiques Supérieure in Montréal (SMS), which was
conceived and supported as a NATO Advanced Study Institute. The courses took
placeduringthetwo-weekperiodfromJune18toJune29,2007,attheCentrede
Recherches Mathématiques (CRM), and they were funded by a grant from NATO
and from the ISM, which is the combined graduate mathematics program of the
Montréal area. The organising committee for this event was D. Bambusi (Milan),
W.Craig(McMaster),S.Kuksin(EdinburghandParis),andA.Neishtadt(Moscow).
Thereweremorethan80participants,comingfromaroundtheworld,andinpartic-
ulartherewereagoodnumberofstudentsfromFrance,fromItaly,fromSpain,from
theUnitedStatesandfromCanada.Theprogramoflecturesoccupiedtwocomplete
weeks, with five or six one-hour lectures each day, so that in total 57 h of courses
werepresented.
Thetopicofthe2007NATO-ASIwasHamiltoniandynamicalsystemsandtheir
applications,whichconcernsmathematicalproblemscomingfromphysicalandme-
chanical systems of evolution equations. Many aspects of the modern theory of
thesubjectwerecovered;topicsoftheprincipallecturesincludedlowdimensional
problemsaswellasthetheoryofHamiltoniansystemsininfinitedimensionalphase
space, and and their applications to problems in classical mechanics, continuum
mechanics, and partial differential equations. Applications were also presented to
several important areas of research, including to celestial mechanics, control the-
ory,thepartialdifferentialequationsoffluiddynamics,andthetheoryofadiabatic
invariants.
Itisagoodthingtodotoarticulatetherelevanceofthesubjectmatterofthese
SMSlecturestothephysicalsciences.Physicallawsareforthemostpartexpressed
in terms of differential equations, and the most natural classes of these are in the
formofconservationlawsorofproblemsofthecalculusofvariationsforanaction
functional. These problems can often be posed as Hamiltonian systems, whether
dynamicalsystemsonfinitedimensionalphasespaceasinclassicalmechanics,or
partial differential equations (PDE) which arenaturally of infinitely many degrees
offreedom.Forinstance,thewellknownN-bodyproblemofcelestialmechanicsis
v
vi Preface
stillofgreatrelevancetomodernmathematicsandmorebroadlytoscience;indeed
in applications the mission design of interplanetary exploration regularly uses the
gravitationalboostofcloseencounterstomanoeuvretheirspacecraft(firstusedin
theMariner–10mission,1974).Thisisalsotrueontheleveloftheoreticalresults,
which can be traced to the work of Laplace, Lagrange and Poincaré, but whose
modernsuccessesdatetothecelebratedtheoryofKolmogorov,ArnoldandMoser
(KAM) (1954/1961/1963). Recent mathematical progress includes the discoveries
ofnewchoreographiesofmanybodyorbits(Chenciner&Montgomery,2000),and
the constructions of Poincaré’s second species orbits (Bolotin & MacKay, 2001).
Furthermore,thedevelopmentofrigorousaveragingmethods(Nekhoroshev1979)
giveshopeforrealisticlongtimestabilityresults(Neishtadt1981,Treschev1996,
Pöschel 1999). Additionally, the last several years has seen major progress in the
long outstanding problem of Arnold diffusion, with the advent of Mather’s varia-
tionaltechniques(2003)relatedtoageneralisedMorse–Hedlundtheory,including
Cheng’ssubsequentworkonvariationalmethods,andthegeometricalapproachto
the‘gapproblem’duetodelaLlave,Delshams&Seara(2006).
Over the last decade the field of Hamiltonian systems has taken on completely
newdirectionsintheextensionoftheanalyticalmethodsofHamiltonianmechan-
ics to partial differential equations. The results of Kuksin, Wayne, Pöschel, Craig,
BambusiandBourgainhaveintroducedanewparadigmtothestudyofpartialdiffer-
entialequationsofevolution,whereresearchfocusesonthefundamentalstructures
invariantunderthedynamicsofthePDEinanappropriatephasespaceoffunctions.
Twobasicexamplesofthisdirectionofenquiryinclude(i)thedevelopmentofsev-
eralapproachestoaKAMtheory,withveryrecentcontributionsbyYuan(2006)and
Eliasson & Kuksin (2007), and (ii) Nekhoroshev stability results for systems with
infinitelymanydegreesoffreedom(Bambusi1999).Theseconsiderationsshowan
excitingandextremelypromisingconnectionbetweenHamiltoniandynamicalsys-
temsandharmonicanalysistechniquesinPDE.Acaseinpointistherelationship
betweenupperboundsonthegrowthofhigherSobolevnormsofsolutionsofnon-
linearevolutionequations,andtheboundsonorbitsgivenbyNekhoroshevtheory;
similarly there is a possibly surprising connection between lower bounds on such
growth and the existence of solution of PDE which exhibit phenomena related to
Arnold diffusion. This research area of evolution equations and Hamiltonian sys-
temsisoneofthemostactiveandexcitingfieldsofPDEinthelastseveralyears.
Thesubjectsinquestioninvolvebynecessitysomeofthemosttechnicalaspects
ofanalysiscomingfromanumberofdiversefields,andbeforeoureventtherehas
not been one venue nor one course of study in which advanced students or oth-
erwise interested researchers can obtain an overview and sufficient background to
enter the field. What we have done with the Montréal Advanced Studies Institute
2007 is to offer a series of lectures encompassing this wide spectrum of topics in
PDEanddynamicalsystems.Mostofthemajordevelopersinthisfieldwerespeak-
ersatthisASI,includingthetopinternationalleadersinthesubject.Thishasmade
it a unique opportunity for junior mathematicians to hear a focused set of lectures
givenbymajorresearchersandcontributorstothefield.Theorganizersaregrateful
for the time and energy that the speakers devoted to the thoughtful preparation of
Preface vii
their lectures, and to the subsequent written and complete versions that appear in
this volume. And in addition the students at this ASI, who were for the most part
advanced graduate students and postdoctoral fellows, included many very promis-
ing and active young mathematicians in the field, with their own well-developed
research programs. The participants’ enthusiasm for the ASI, their help in writing
lecturenotesforthecourses,andtheirgeneralcheerfulnessandgoodattitudedur-
ingthecourseofthetwoweeksoflectures,madetheeventanexperiencenottobe
forgotten.
Last but not least, the organizers of the SMS 2007 would like to acknowledge
thegenerousandtimelysupportofthePublicDiplomacyDivisionofNATO,with-
out which the two weeks of this Advanced Study Institute would not have taken
place,theadditionalfinancialsupportoftheMontréalCentredeRecherchesMath-
ématiques (CRM), the ISM and the Université de Montréal, and for the depend-
ableguidanceandinitiativeofSakinaBenhima,ourDirectricedeProgrammeatthe
CRMinMontréal.
Theseriesoflecturesinthisvolumeincludesthefollowingtopics:Hamiltonian
systemsandoptimalcontrol(A.Agrachev,SISSA,Trieste),Birkhoffnormalform
for some semilinear PDEs (D. Bambusi, Universita degli Studi di Milano), Varia-
tionalmethodsforHamiltonianPDEs(M.Berti,UniversitàdegliStudidiNapoli),
The N-body problem (A. Chenciner, Observatoire de Paris), Variational methods
fortheproblemofArnolddiffusion(C.-Q.Cheng,NanjingUniversity),Thetrans-
formation theory of Hamiltonian PDE and the problem of water waves (W. Craig,
McMasterUniversity),Geometricapproachestodiffusionandinstability(R.dela
Llave,UniversityofTexasatAustin),KAMforthenonlinearSchrödingerequation
(H.Eliasson,UniversitédeParis7),GroupsandtopologyinEulerhydrodynamics
andtheKdV(B.Khesin,UniversityofToronto),ThreetheoremsonperturbedKdV
(S. Kuksin, Heriot-Watt University), Averaging methods and adiabatic invariants
(A.I.Neishtadt,SpaceResearchInstitute,RussianAcademyofScience),Periodic
KdV equation in weighted Sobolev spaces (J. Pöschel, Universität Stuttgart), The
forced pendulum as a model for dynamical behavior (P. Rabinowitz, University
of Wisconsin), Normal forms of holomorphic dynamical systems (L. Stolovitch,
UniversitéPaulSabatier),SomeaspectsoffinitedimensionalHamiltoniansystems
(D. Treschev, Moscow State University), Infinite dimensional dynamical systems
andtheNavier–Stokesequations(C.E.Wayne,BostonUniversity),andKAMthe-
orywithapplicationstononlinearwaveequations.(X.Yuan,FudanUniversity).
HamiltonandMontréal,Canada WalterCraig
July2007
Contents
Someaspectsoffinite-dimensionalHamiltoniandynamics ............. 1
D.V.Treschev
1 Symplecticstructure.InvariantformoftheHamiltonianequations 1
1.1 Hamiltonianequations ............................. 1
1.2 ThePoissonbracket ............................... 3
1.3 Liouvilletheoremoncompletelyintegrablesystems .... 4
2 Apendulumwithrapidlyoscillatingsuspentionpoint ........... 5
3 Anti-integrablelimit....................................... 8
3.1 Thestandardmap ................................. 8
3.2 Anti-integrablelimit ............................... 9
3.3 ProofoftheAubrytheorem ......................... 11
3.4 Someremarks .................................... 12
4 Separatrixsplitting ........................................ 13
4.1 Poincaré’sobservation ............................. 13
4.2 ThePoincaréintegral .............................. 14
4.3 ProofofTheorem5................................ 15
4.4 Standardexample ................................. 18
References..................................................... 19
FourlecturesontheN-bodyproblem............................... 21
AlainChenciner
1 ThePoincaré–Birkhoff–Conleytwistmapoftheannulus
fortheplanarcircularrestrictedthree-bodyproblem ............ 21
1.1 TheKeplerproblemasanoscillator .................. 21
1.2 Therestrictedprobleminthelunarcase............... 22
1.3 Hill’ssolutions.................................... 24
1.4 Theannulustwistmap ............................. 26
2 The Arnold–Herman stability theorem for the spatial
(1+n)-bodyproblem ...................................... 29
2.1 ThesecularHamiltonian............................ 30
2.2 Herman’snormalformtheoremandhowtouseit....... 33
ix
x Contents
2.3 Astabilitytheorem ................................ 35
2.4 Herman’sdegeneracy .............................. 36
3 MinimalactionandMarchal’stheorem ....................... 37
3.1 Centralconfigurationsandtheirhomographicmotions... 37
3.2 VariationalcharacterizationsofLagrange’sequilateral
solutions......................................... 38
3.3 Marchal’stheorem................................. 40
3.4 Minimizationundersymmetryconstraints ............. 42
4 Globalcontinuationviaminimization ........................ 43
4.1 BifurcationsfromtheLagrangeequilateralrelative
equilibrium....................................... 44
4.2 FromtheequilateraltriangletotheEight .............. 46
4.3 FromthesquaretotheHip-Hop ..................... 48
4.4 Theavatarsoftheregularn-gonrelativeequilibrium:
eights,chainsandgeneralizedHip-Hops .............. 50
References..................................................... 50
Averagingmethodandadiabaticinvariants.......................... 53
AnatolyNeishtadt
1 Introduction.............................................. 53
2 Adiabaticinvarianceinone-frequencysystems................. 54
3 Onadiabaticinvarianceinmulti-frequencysystems............. 63
References..................................................... 65
Transformation theory of Hamiltonian PDE and the problem
ofwaterwaves .................................................. 67
WalterCraig
1 Hamiltoniansystems ...................................... 67
2 PartialdifferentialequationsasHamiltoniansystems............ 68
3 Theproblemofwaterwaves ................................ 71
4 TheDirichlet–Neumannoperator ............................ 73
5 Perturbationtheory........................................ 74
6 Thecalculusoftransformations ............................. 75
References..................................................... 82
ThreetheoremsonperturbedKdV................................. 85
SergeiB.Kuksin
1 KdVequation ............................................ 85
1.1 Integrabilityof(KdV).............................. 86
1.2 Normalformsfor(KdV)............................ 87
2 KAM-theory ............................................. 88
3 Averaging:Hamiltonianperturbations ........................ 89
4 Averaging:caseofnon-Hamiltonianperturbations.............. 90
4.1 Deterministicperturbations ......................... 90
4.2 Randomperturbations.............................. 90
References..................................................... 91