some bi-hamiltonian equations in $r^3$new

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1、HAMILTONIANEQUATIONSINR3AhmetAy,MetinG¨ursesandKostyantynZheltukhinDepartmentofMathematics,FacultyofSciencesBilkentUniversity,06800Ankara,TurkeyFebruary5,2008AbstractHamiltonianformulationofN=3systemsisconsideredingeneral.ThemostgeneralsolutionoftheJacobiequationinR3is

2、proposed.Theformofthesolutionisshowntobevalidalsointheneighbor-hoodofsomeirregularpoints.CompatiblePoissonstructuresandcorrespondingbi-Hamiltoniansystemsarealsodiscussed.Hamilto-nianstructures,classificationofirregularpointsandthecorrespond-arXiv:nlin/0304002v3[nlin.SI]

3、26Aug2003ingreducedfirstorderdifferentialequationsofseveralexamplesaregiven.01.Introduction.Hamiltonianformulationofasystemofdynamicalequationsisimportantnotonlyinmathematicsbutalsoinphysicsandotherbranchesofnaturalsciences.Theyingeneraldescribeconservedsystems.Amongallp

4、ossibleodddimensionalcasesthethreedimensionaldynamicalsystemshaveauniqueposition.TheJacobiequationinthiscasereducestoasinglescalarequationforthreecomponentsofthePoissonstructureJ.DuetothispropertyN=3dynamicalsystemsattractedmanyresearchestoderivenewHamiltoniansystems,[

5、6]–[12].Morerecently[1],[2]alargeclassofsolutionsoftheJacobiequationinR3wasgiven.Poissonstructures,inalldimensions,werealsoconsideredin[3].Inthiswork,weconsidergeneralsolutionofJacobiequationinR3.WefindthecompatiblePoissonstructuresandgivethecorrespondingbi-Hamiltonians

6、ystems.Wegiveallexplicitexamplesinaspecialsectionandatableattheend.LetusgivenecessaryinformationaboutthePoissonstructuresinR3.AmatrixJ=(J),i,j=1,2,3,definesaPoissonstructureinR3ifitisijskew-symmetric,Jij=−Jji,anditsentriessatisfytheJacobiequationlijkljkilkijJ∂lJ+J∂lJ+J∂

7、lJ=0,(1)wherei,j,k=1,2,3.Hereweusethesummationconvention,meaningthatrepeatedindicesaresummedup.Letusintroducethefollowingnotations.FormatrixJputJ12=u,J31=v,J23=w.ThenJacobiequation(1)takes1theformu∂1v−v∂1u+w∂2u−u∂2w+v∂3w−w∂3v=0.(2)Itcanalsoberewrittenas2v2u2wu∂1+w∂2+v∂

8、3=0.(3)uwv(Weassumethatnoneofthefunctionsu,vandwvanish.Ifanyoneofthesefunctionsvanishesthentheequation(2)becomestrivi