SLIM — a formalism for linear coupled systems

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Courant E D,Snyder H S.Ann.Phys.,1958,3:1[2]There are many such examples:MomentumMomentum compaction ac=1/C∮ηds/pClosed orbit Δx=θk√βkβ/2sinπvxcos[πvx-|ψ-ψk|]Beam size σ2x=σ2xβ+σ2δη2σ2xβ/β=55/32√3h/mcγ2/1-D∮Hds/|p|3/∮ds/p2 Damping parti-Jx=1-D,Jy=1,Js=2+D tion D=∮ηds/p(2K+1/p2)/∮ds/p2[3]Alexander W.Chao,J.Appl.Phys.,1979,50(2):595[4]Alexander W.Chao.Nucl.Instrum.Methods,1981,180:29[5]Edwards D A,Teng L.IEEE Trans Nucl.Sci.,1973,20:3[6]Ruggiero F,Picnsso E,Radicati L.Ann.Phys.,1990,197:439[7]Barber D,Heinemann K,Mais H,Ripkin G.DESY-91-146,1991[8]Ohmi K,Hirata K,Oide K.Phys.Rev.E,1994,49;751[9]Forest E.Phys.Rev.E,1998,58:2481[10]Wolski A.Phys.Rev.ST Accel.Beams,2006,9:024001[11]Nash B.Ph.D.Thesis,Stanford University,2006[12]This is slightly exaggerated.The calculation as given here is valid only if the tunes stay away from the resonances by a distance of the order of the radiation damping constants.Since the radiation damping time constants are typically extremely small,the sacrifice of parameter space is negligibly small