Multiresolution Representation for Orbital Dynamics in Multi

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.