Fractional Variations for Dynamical Systems Hamilt

arXiv:math-ph/0606048v120Jun2006JournalofPhysicsA.Vol.39.No.26.(2006)pp.8409-8425.FractionalVariationsforDynamicalSystems:HamiltonandLagrangeApproachesVasilyE.TarasovSkobeltsynInstituteofNuclearPhysics,MoscowStateUniversity,Moscow119992,RussiaE-mail:[e-mail]ralizationofanexteriorderivativeforcalculusofvariationsisdefined.TheHamiltonandLagrangeapproachesareconsidered.FractionalHamiltonandEuler-Lagrangeequationsarederived.FractionalequationsofmotionareobtainedbyfractionalvariationofLagrangianandHamiltonianthathaveonlyintegerderivatives.PACSnumbers:45.20.-d;45.20.Jj;45.10.Hj1IntroductionThetheoryofderivativesofnonintegerorder[1,2]goesbacktoLeibniz,Liouville,Riemann,GrunwaldandLetnikov[2].Derivativesandintegralsoffractionalorderhavefoundmanyapplicationsinrecentstudiesinmechanicsandphysics.Inafairlyshortperiodoftimethelistofsuchapplicationsbecomeslong.Forexample,itincludeschaoticdynamics[3,4],mechanicsoffractalmedia[5,6,7,8],quantummechanics[9,10],physicalkinetics[3,11,12,13,14,15,16],plasmaphysics[17,18],astrophysics[19],long-rangedissipation[20,21],mechanicsofnon-Hamiltoniansystems[22,23],theoryoflong-rangeinteraction[24,25,26],anomalousdiffusionandtransporttheory[3,27,28]andmanyothersphysicaltopics.Inmathematicsandtheoreticalphysics,thevariational(functional)derivativeisageneral-izationoftheusualderivativethatarisesinthecalculusofvariations.Inavariationinsteadofdifferentiatingafunctionwithrespecttoavariable,onedifferentiatesafunctionalwithrespecttoafunction.Inthispaper,weconsiderthefractionalgeneralizationofvariational(functional)exteriorderivatives.Themainresultsarederivedinsections4.2,5.2,6.2,and6.3.Insections2,3,4.1,5.1,6.1,briefreviewsoffractionalderivatives,differentialforms,Hamiltoniansystemsareconsideredtofixnotationandprovideconvenientreferences.Insection2,abriefreviewofdifferentialformsisconsidered.Insection3,weconsiderHamiltonianandfractionalHamiltoniansystems[23].Insection4,wedefinethefractionalvariationsinHamilton’sapproachtodescribethemotion.Thefractionalgeneralizationofstationaryactionprincipleissuggested.Insection5,wediscussthefractionalvariationsinLagrange’sapproachtodescribethemotion,andthefractionalgeneralizationofstationaryactionprincipleissuggested.Insection6,weconsiderthegeneralizationofactionprincipletonon-Hamiltoniansystems.Thefractionalequationsofmotionwithfrictionarediscussed.Finally,ashortconclusionisgiveninsection7.12FractionalDerivativesandDifferentialForms2.1DifferentialFormsInthissubsection,abriefreviewofdifferentialforms[29,30]isconsideredtofixnotationandprovideaconvenientreference.Definition1.Adifferential1-formω=Fi(x)dxi(1)iscalledanexact1-forminRnifthevectorfieldFi(x)canbepresentedasFi(x)=−∂V∂xi,(2)whereV=V(x)isacontinuouslydifferentiablefunction.Inthiscase,thedifferentialform(1)isanexactformω=−dV,whereV=V(x)isacontinuouslydifferentiablefunction(0-form).Heredistheexteriorderivative[29].TheexteriorderivativeofthefunctionVisthe1-formdV=dxi∂V/∂xiwritteninacoordinatechart(x1,...,xn).Forthek-formωkandthel-formωl,theexteriorderivativeobeystherelationd(ωk∧ωl)=(dωk)∧ωl+(−1)kωk∧dωl.(3)Herekandlareintegers.Notethatddω=0foranyformω.Ifdω=0,thenωiscalledaclosedform.Inmathematics[29],theconceptsofclosedandexactformsaredefinedbytheequationdω=0foragivenωtobeaclosedform,andω=dhforanexactform.Itisknownthattobeexactisasufficientconditiontobeclosed.Inabstracttermsthequestionofwhetherthisisalsoanecessaryconditionisawayofdetectingtopologicalinformationbydifferentialconditions.Proposition1.IfasmoothvectorfieldF=eiFi(x)satisfiestherelations∂Fi∂xj−∂Fj∂xi=0(4)onacontractibleopensubsetXofRn,then(1)istheexactformsuchthatω=−∂V(x)∂xidxi.(5)Proof.Letusconsidertheforms(1).Theformulafortheexteriorderivativeof(1)isdω=12∂Fi∂xj−∂Fj∂xi!dxj∧dxi,where∧isthewedgeproduct[29].Thereforetheconditionforωtobeclosedis(4).IfFi=−∂V/∂xi,thentheimplicationfrom’exact’to’closed’isaconsequenceoftheper-mutabilityofthesecondderivatives.ForthesmoothfunctionV=V(x),thesecondderivativecommute,andequation(4)holds.22.2FractionalDifferentialFormsIfthepartialderivativesinthedefinitionoftheexteriorderivatived=dxi∂∂xiareallowedtoassumefractionalorder,thenafractionalexteriorderivativeisdefined[31]bydα=(dxi)αDαxi.(6)HereweuseDαxf(x)=1Γ(m−α)Zx0dy(x−y)α−m+1∂mf(y)∂ym,(7)whereα0,andmisthefirstwholenumbergreaterthanorequaltoα.Equation(7)definestheCaputofractionalderivatives[33,34,5,35]oforderα0.Definition2.Adifferential1-formωα=Fi(x)(dxi)α(8)iscalledanexactfractionalformifthethevectorfieldFi(x)canberepresentedasFi(x)=−DαxiV,(9)whereV=V(x)isacontinuouslydifferentiablefunction,andDαxiisaderivativeoforderα.Using(6)theexactfractionalformcanberepresentedasωα=−dαV=−(dxi)αDαxiV.Therefore,wehave(9).Notethatequation(8)isafractionalgeneralizationofthedifferentialform(1).Obviouslythatfractional1-formωαcanbeclosedwhenthedifferential1-formω=ω1isnotclosed.Thefractionalanalogueofproposition1hastheform.Proposition2.IfasmoothvectorfieldF=eiFi(x)onacontractibleopensubsetXofRnsatisfiestherelationsDαxjFi−DαxiFj=0,(10)thentheform(8)isanexactfractional1-formsuchth