Multiresolution Representation for Orbital Dynamics in Multi(2)

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)