Multiresolution Representation for Orbital Dynamics in Multi(3)

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