Differential equations - geometry, symmetries and integrability: The Abel symposium 2008

393 Pages·2009·4.88 MB·English

by  R. J. Alonso

ABEL SYMPOSIA
Edited by the Norwegian Mathematical Society
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Boris Kruglikov (cid:129) Valentin Lychagin
Eldar Straume
Editors
Differential Equations:
Geometry, Symmetries
and Integrability
The Abel Symposium 2008
Proceedings of the Fifth Abel Symposium,
Tromsø, Norway, June 17-22, 2008
Editors
BorisKruglikov EldarStraume
ValentinLychagin DepartmentofMathematics
DepartmentofMathematics NorwegianUniversityofScience
andStatistics andTechnology
FacultyofScience 7034Trondheim
UniversityofTromsø Norway
9037Tromsø [email protected]
Norway
[email protected]
[email protected]
ISBN978-3-642-00872-6 e-ISBN978-3-642-00873-3
DOI10.1007/978-3-642-00873-3
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2009931139
Mathematics Subject Classificatio (2000): 58-06, 35A30, 34C14, 37K05, 35Q51, 53A55, 35A27,
58H05,53D50
(cid:2)c Springer-VerlagBerlinHeidelberg2009
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Preface to the Series
The Niels Henrik Abel Memorial Fund was established by the Norwegian
governmentonJanuary1.2002.ThemainobjectiveistohonorthegreatNorwegian
mathematicianNielsHenrikAbelbyawardinganinternationalprizeforoutstanding
scientificworkinthefieldofmathematics.Theprizeshallcontributetowardsrais-
ingthestatusofmathematicsinsocietyandstimulatetheinterestforscienceamong
school children and students. In keeping with this objective the board of the Abel
fundhasdecidedtofinanceoneortwoAbelSymposiaeachyear.Thetopicmaybe
selectedbroadlyintheareaofpureandappliedmathematics.TheSymposiashould
beatthehighestinternationallevel,andservetobuildbridgesbetweenthenational
and international research communities. The Norwegian Mathematical Society is
responsiblefortheevents.Ithasalsobeendecidedthatthecontributionsfromthese
Symposia should be presented in a series of proceedings, and Springer Verlag has
enthusiastically agreed to publish the series. The board of the Niels Henrik Abel
Memorial Fund is confident that the series will be a valuable contribution to the
mathematicalliterature.
RagnarWinther
ChairmanoftheboardoftheNielsHenrikAbelMemorialFund
v
Preface
TheAbelSymposium2008focusedonthemoderntheoryofdifferentialequations
andtheirapplicationsingeometry,mechanics,andmathematicalphysics.Following
the tradition of Monge, Abel and Lie, the scientific program emphasized the role
ofalgebro-geometricmethods,whichnowadayspermeateallmathematicalmodels
in natural and engineering sciences. The ideas of invariance and symmetry are of
fundamentalimportanceinthegeometricapproachtodifferentialequations,witha
seriousimpactcomingfromtheareaofintegrablesystemsandfieldtheories.More
specifically,thefollowingtopicswerecentraltotheSymposium:
• Integrabilitymethodsforlinearandnon-lineardifferentialequations
• Geometricanalysisofthesolutionsandtheirmodulispaces
• Symmetries,conservationlaws,recursionoperatorsandgeneralizations
• Lieequationsandgeometricstructures
• Pseudogroupactionsandthealgebraofdifferentialinvariants
• TopicsfromHamiltonianandcelestialmechanics
• Differentialequationswithinfieldtheoriesandquantumphysics.
TheSymposiumwashostedbytheUniversityofTromsø,theworld’snorthernmost
university,fromJune17–22,2008,andwasorganizedby:
• IanAnderson,UtahStateUniversity
• AlainChenciner,Universite´ ParisVIIandIMCCE
• BorisKruglikov,UniversityofTromsø
• DemeterKrupka,PalackyUniversity
• ValentinLychagin,UniversityofTromsø
• EldarStraume,NTNU-Trondheim
Leadingresearchersfrom13differentcountries,workingintheseareaswereinvited
toparticipate.Atotalof43participantswereinvited,includingfivePh.D.students
andpost-doctoralresearchersfromNorway.Thedailyprogramconsistedoflectures
and discussions, held in an open and encouraging atmosphere. The following 28
plenarylecturesweregiven:
vii
viii Preface
1. Alain Albouy: Projective dynamics of a classical particle or a multiparticle
system,
2. IanAnderson:SymmetryreductionandDarbouxintegrability,
3. Ugo Bruzzo: Framed bundles on Hirzebruch surfaces and equivariant coho-
mologyofmodulispaces,
4. AndreasCap:BGGsequencesandgeometricoverdeterminedsystems,
5. Alain Chenciner: Action minimization and global continuation of Lyapunov
familiesstemmingfromrelativeequilibria,
6. PhilippeDelanoe:Transportequationsandgeometry:asurvey,
7. BorisDubrovin:Ondeformationsofintegrablehierarchies,
8. VladislavGoldberg(togetherwithV.Lychagin):Abelianequationsanddiffer-
entialinvariantsofplanar4-webs,
9. MichaelHazewinkel:Nicenesstheorems,
10. Wu-Yi Hsiang: On the critical optimality for sphere packings of mixtures of
twosizes,
11. NailIbragimov:Symmetriesandconservationlaws:ageneraltheoremforar-
bitrarydifferentialequations,
12. NikyKamran:FocalsystemsforPfaffiansystemswithcharacteristics,
13. BorisKruglikov(togetherwithV.Lychagin):Algebraicaspectsofthecompat-
ibilityofPDEs,
14. DemeterKrupka:Differentialinvariantsingaugetheory,
15. BorisKupershmidt:Phasespacesinalgebra,
16. BernardMalgrange:LiepseudogroupsanddifferentialGaloistheory,
17. AndreiMarshakov:Gauge/stringduality,integrableequationsandAbeliandif-
ferentials,
18. JohnMather:TonelliminimizersandrelativeTonelliminimizers,
19. Vladimir Matveev: Projective vector fields: Solution of Lie and Schouten
problems,
20. SergeiMerkulov:Wheeledprops,deformationtheoryandquantization,
21. RichardMoeckel:Topicsonthethree-bodyproblem,
22. JuanMorales-Ruiz(togetherwithD.Blazquez):Localandglobalpropertiesof
Lie-Vessiotsystems,
23. JesuRodriguez:CharacteristicsofPDEintheframeworkofLie-Weiljetspaces,
24. VladimirRoubtsov:PartitionfunctionofSOSellipticmodelswithdomainwall
boundaryconditionsandprojectionmethod,
25. PerTomter:Isometricimmersionsofhomogeneoushyperspheresintocomplex
hyperbolicspace,
26. Alexander Verbovetsky (together with P. Kersten, J. Krasilshchik and
R.Vitolo):HamiltonianstructuresforgeneralPDEs,
27. Raffaele Vitolo (together with M. Modugno and C. Tejero Prieto): Geometric
aspectsofthequantizationofarigidbody,
28. KeizoYamaguchi:Contactgeometryofsecondorder.
Preface ix
Inadditiontotheabovespeakers(andorganizers),thefollowinginvitedguestsalso
attendedthesymposium:
1. DavidBlazquez,Barcelona
2. TorFla˚,Tromsø
3. HiljaLisaHuru,Oslo
4. MarteHøyem,Tromsø
5. PerJakobsen,Tromsø
6. CathrineJensen,Tromsø
7. JosephKrasilshchik,Moscow
8. AlexeyKushner,Astrakhan
9. EinarMjølhus,Tromsø
10. MartinRypdal,Tromsø
11. MahdiKhajehSalehani,Trondheim
12. AlexeySamokhin,Moscow
13. LarsSydnes,Trondheim
Wearegratefultoalltheparticipantsfortheirvaluablecontributionsandformaking
the symposium a successful event. We would also like to express our gratitude to
the Abel Foundation and the Norwegian Mathematical Society for giving us the
opportunitytoarrangetheAbelSymposium2008.
Moreover,wewouldliketothanktheUniversityofTromsøforprovidinguswith
additionalsupportandpracticalassistancerelatedtothepreparationandorganiza-
tionofthesymposium.
TromsøandTrondheim BorisKruglikov
January28,2009 ValentinLychagin
EldarStraume
Contents
Some Canonical Structures of Cartan Planes
inJetSpacesandApplications................................................ 1
RicardoJ.Alonso,SoniaJime´nez,andJesu´sRodr´ıguez
TransformationsofDarbouxIntegrableSystems............................ 21
IanM.AndersonandMarkE.Fels
DifferentialGeometricHeuristicsforRiemannianOptimal
MassTransportation .......................................................... 49
PhilippeDelanoe¨
OnRankProblemsforPlanarWebsandProjectiveStructures............. 75
VladislavV.GoldbergandValentinV.Lychagin
NicenessTheorems ............................................................107
MichielHazewinkel
ThePolynomialAlgebraandQuantizationsofElectromagnetic
Fields...........................................................................151
HiljaL.Huru
ABridgeBetweenLieSymmetriesandGaloisGroups......................159
NailH.Ibragimov
FocalSystemsforPfaffianSystemswithCharacteristics....................173
NikyKamran
HamiltonianStructuresforGeneralPDEs...................................187
(cid:2)
P.Kersten,I.S.Krasilshchik,A.M.Verbovetsky,andR.Vitolo
Point Classification of Second Order ODEs: Tresse
ClassificationRevisitedandBeyond..........................................199
BorisKruglikov
ClassificationofMonge–Ampe`reEquations..................................223
AlexeiG.Kushner
xi
xii Contents
OnNonabelianTheoriesandAbelianDifferentials..........................257
A.Marshakov
GeometricAspectsoftheQuantizationofaRigidBody.....................275
M.Modugno,C.TejeroPrieto,andR.Vitolo
Shooting for the Eight: A Topological Existence Proof
foraFigure-EightOrbitoftheThree-BodyProblem........................287
RichardMoeckel
CompatiblePoissonBrackets,QuadraticPoissonAlgebrasand
Classicalr-Matrices ...........................................................311
V.RoubtsovandT.Skrypnyk
ContactGeometryofSecondOrderI ........................................335
KeizoYamaguchi