Multiresolution Representation for Orbital Dynamics in Multipolar Fields

We present the applications of variation -- wavelet analysis to polynomial/rational approximations for orbital motion in transverse plane for a single particle in a circular magnetic lattice in case when we take into account multipolar expansion up to an a

MULTIRESOLUTIONREPRESENTATIONFORORBITALDYNAMICSIN

MULTIPOLARFIELDS

000

A.Fedorova,M.Zeitlin,IPME,RAS,V.O.Bolshojpr.,61,199178,St.Petersburg,Russia 2 guAbstract

AWepresenttheapplicationsofvariation–waveletanalysis 3topolynomial/rationalapproximationsfororbitalmotionin1transverseplaneforasingleparticleinacircularmagnetic latticeincasewhenwetakeintoaccountmultipolarexpan-]sionuptoanarbitrary nitenumberandadditionalkickphterms.Wereduceinitialdynamicalproblemtothe nite-cnumber(equaltothenumberofn-poles)ofstandardalge-cbraicalproblems.Wehavethesolutionasamultiresolutiona(multiscales)expansioninthebaseofcompactlysupported.swaveletbasis.

cisy1INTRODUCTION

hpInthispaperweconsidertheapplicationsofanewnumeri-[cal-analyticaltechniquewhichisbasedonthemethodsof localnonlinearharmonicanalysisorwaveletanalysistothe1orbitalmotionintransverseplaneforasingleparticleinav5circularmagneticlatticeincasewhenwetakeintoaccount4multipolarexpansionuptoanarbitrary nitenumberand0additionalkickterms.Wereduceinitialdynamicalprob-8lemtothe nitenumber(equaltothenumberofn-poles)of0standardalgebraicalproblemsandrepresentalldynamical0variablesasexpansioninthebasesofmaximallylocalized/0inphasespacefunctions(waveletbases).Waveletanalysisscisarelativelynovelsetofmathematicalmethods,whichigivesusapossibilitytoworkwithwell-localizedbasesinsyfunctionalspacesandgivesforthegeneraltypeofopera-htors(differential,integral,pseudodifferential)insuchbasespthemaximumsparseforms.Ourapproachinthispaperis:vbasedonthegeneralizationofvariational-waveletapproachiX

from[1]-[8],whichallowsustoconsidernotonlypolyno-mialbutrationaltypeofnonlinearities[9].Thesolutionr

ahasthefollowingform

z(t)=zslow

N(t)+

zj(ωjt),ωj～2j(1)

j≥N

whichcorrespondstothefullmultiresolutionexpansionin

alltimescales.Formulagivesusexpansionintoaslow

partzslow

NandfastoscillatingpartsforarbitraryN.So,wemaymovefromcoarsescalesofresolutiontothe nestoneforobtainingmoredetailedinformationaboutourdynami-calprocess.The rsttermintheRHSofequation(1)corre-spondsonthegloballeveloffunctionspacedecompositiontoresolutionspaceandthesecondonetodetailspace.Inthiswaywegivecontributiontoourfullsolutionfromeachscaleofresolutionoreachtimescale.Thesameiscorrect

+

2

1

y2

2+k1(s)

(n+1)!

·(x+iy)(n+1)

Thenwemaytakeintoaccountarbitrarybut nitenumberoftermsinexpansionofRHSofHamiltonianandfromourpointofviewthecorrespondingHamiltonianequationsofmotionsarenotmorethannonlinearordinarydifferen-tialequationswithpolynomialnonlinearitiesandvariablecoef cients.Alsowemayaddthetermscorrespondingtokicktypecontributionsofrf-cavity:

Aτ=

L

L

τ

·δ(s s0)

(5)

orlocalizeds0)= cavityV(s)=V=+∞

0·δp(s s0)withδp(s

nn= ∞δ(s (s0+n·L))atpositions0.Fig.1andFig.2present nitekicktermmodelandthecorrespondingmultiresolutionrepresentationoneachlevelofresolution.

We present the applications of variation -- wavelet analysis to polynomial/rational approximations for orbital motion in transverse plane for a single particle in a circular magnetic lattice in case when we take into account multipolar expansion up to an a

Figure1:Finitekickmodel.

Figure2:Multiresolutionrepresentationofkick.

3RATIONALDYNAMICS

The rstmainpartofourconsiderationissomevariational

approachtothisproblem,whichreducesinitialproblemtotheproblemofsolutionoffunctionalequationsatthe rststageandsomealgebraicalproblemsatthesecondstage.Wehavethesolutioninacompactlysupportedwaveletba-sis.Multiresolutionexpansionisthesecondmainpartofourconstruction.Thesolutionisparameterizedbysolu-tionsoftworeducedalgebraicalproblems,oneisnonlin-earandthesecondaresomelinearproblems,whichareobtainedfromoneofthenextwaveletconstructions:themethodofConnectionCoef cients(CC),StationarySub-divisionSchemes(SSS).

3.1VariationalMethod

Ourproblemsmaybeformulatedasthesystemsofordi-narydifferentialequations

Qi(x)

dxi

dt

(Qiyi)+Piyi(7)

andasetoffunctionals

x)= 1

Fi(Φi(t)dt Qixiyi|10,

(8)

whereyi(t)(yi(0)=0)aredual(variational)variables.It

isobviousthattheinitialsystemandthesystem

Fi(x)=0

(9)

areequivalent.Ofcourse,weconsidersuchQi(x)whichdonotleadtothesingularproblemwithQi(x),whent=0ort=1,i.e.Qi(x(0)),Qi(x(1))=∞.

Nowweconsiderformalexpansionsforxi,yi:

xi(t)=xi(0)+ λki k(t)yj(t)=

ηr

j r(t),(10)

k

r

where k(t)areusefulbasisfunctionsofsomefunctionalspace(L2,Lp,Sobolev,etc)correspondingtoconcreteproblemandbecauseofinitialconditionsweneedonly k(0)=0,r=1,...,N,i=1,...,n,

λ={λi}={λri}=(λ1i,λ2i,...,λN

i),

(11)

wherethelowerindexicorrespondstoexpansionofdy-namicalvariablewithindexi,i.e.xiandtheupperindexrcorrespondstothenumbersoftermsintheexpansionofdynamicalvariablesintheformalseries.Thenweput(10)intothefunctionalequations(9)andasresultwehavethefollowingreducedalgebraicalsystemofequationsonthesetofunknowncoef cientsλkiofexpansions(10):

L(Qij,λ,αI)=M(Pij,λ,βJ),

(12)

whereoperatorsLandMarealgebraizationofRHSandLHSofinitialproblem(6),whereλ(11)areunknownsofreducedsystemofalgebraicalequations(RSAE)(12).

Qijarecoef cients(withpossibletimedependence)ofLHSofinitialsystemofdifferentialequations(6)andasconsequencearecoef cientsofRSAE.

Pijarecoef cients(withpossibletimedependence)ofRHSofinitialsystemofdifferentialequations(6)andasconsequencearecoef cientsofRSAE.

I=(i1,...,iq+2),J=(j1,...,jp+1)aremultiindexes,bywhicharelabelledαIandβI—othercoef cientsofRSAE(12):

βJ={βj1...jp+1}=

jk,(13)1≤jk≤p+1

wherepisthedegreeofpolinomialoperatorP(6)

αI={αi1...αiq+2}=

i1,...,i q+2

i1... ˙is... iq+2,(14)

We present the applications of variation -- wavelet analysis to polynomial/rational approximations for orbital motion in transverse plane for a single particle in a circular magnetic lattice in case when we take into account multipolar expansion up to an a

whereqisthedegreeofpolynomialoperatorQ(6),i =(1,...,q+2), ˙is=d is/dt.

Now,whenwesolveRSAE(12)anddetermineunknowncoef cientsfromformalexpansion(10)wethereforeob-tainthesolutionofourinitialproblem.Itshouldbenotedifweconsideronlytruncatedexpansion(10)withNtermsthenwehavefrom(12)thesystemofN×nalgebraicalequationswithdegree =max{p,q}andthedegreeofthisalgebraicalsystemcoincideswithdegreeofinitialdif-ferentialsystem.So,wehavethesolutionoftheinitialnonlinear(rational)problemintheform

xi(t)=xi(0)+

N

λkiXk(t),

(15)

k=1

wherecoef cientsλkiarerootsofthecorrespondingre-ducedalgebraical(polynomial)problemRSAE(12).Con-sequently,wehaveaparametrizationofsolutionofinitial

problembysolutionofreducedalgebraicalproblem(12).The rstmainproblemisaproblemofcomputationsofcoef cientsαI(14),βJ(13)ofreducedalgebraicalsys-tem.Theseproblemsmaybeexplicitlysolvedinwaveletapproach.

Nextweconsidertheconstructionofexplicittimesolu-tionforourproblem.Theobtainedsolutionsaregivenintheform(15),whereXk(t)arebasisfunctionsandλikarerootsofreducedsystemofequations.InourcaseXk(t)areobtainedviamultiresolutionexpansionsandrepresentedbycompactlysupportedwaveletsandλikaretherootsofcorre-spondinggeneralpolynomialsystem(12)withcoef cients,whicharegivenbyCCorSSSconstructions.Accordingtothevariationalmethodtogivethereductionfromdifferen-tialtoalgebraicalsystemofequationsweneedcomputetheobjectsαIandβJ[1],[9].

Ourconstructionsarebasedonmultiresolutionappro-ach.Becauseaf negroupoftranslationanddilationsisinsidetheapproach,thismethodresemblestheactionofamicroscope.Wehavecontributionto nalresultfromeachscaleofresolutionfromthewholein nitescaleofspaces.Moreexactly,theclosedsubspaceVj(j∈Z)correspondstoleveljofresolution,ortoscalej.Weconsideramul-tiresolutionanalysisofL2(Rn)(ofcourse,wemaycon-sideranydifferentfunctionalspace)whichisasequenceofincreasingclosedsubspacesVj:

...V 2 V 1 V0 V1 V2 ...

satisfyingthefollowingproperties:

Vj=0

,

(16)

j∈Z

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