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Additional resources for Appl of Differential Algebra to Single-Particle Dynamics in Storage Rings
F/+ 1, given by Eq. 26), are obtained once li(z-") is obtained. 2 Eigenfunctions of the Linear Normal Form Since (:r-T_-l(z-')) in Eq. 26)is a Lie operator, it follows that we can obtain Fi(_ if we can decompose fi(_ and hi(J) into polynomials of the eigenfunctions of T_-I(_ or T_(_. For simplicity without losing generality, we shall consider a transverse map. Then T_-I(z -') -- e:fi'Y: _. e:p=J=+l_Ju:, 1 where J_ - ½(x 2 + p_) and Jy - ½(y2 + p_). It is clear that _± - :_(x 4- ipx) and _± = 7_(y 1 -4-ipp) are eigenfunctions of the Lie operators • Jx " and • Jy ", respectively, as we have •j.
Tl cos(#0 + 6#a ++ . 12) ' and - 4 det(/Mi) hi = (ai - di)2 + (bi + ci) 2 = - -[Tr(iMi)12 L jsm/_0 . A canonical transformation can then be made to obtain,+1M(6) asfollows: ,+_g(6) = (I- 6'A_)_M(6)(I + _'A,) = (I- 6_A,)(R___(6) + 6_M_)(I + 6_A_) + 6i+1(I -- 6iAi)iMi+l(I +... t,)_ + `5i/_i)] ' and Tr(iMi) _ui --" ai + di = 2 sin/_0 2 sin/_0 • The above process is then iterated until we obtain the nth-order symplectic generation matrix 2,,(,5) -- I + 6nAn and then make the (n + 1)th canonical transformation to obtain (I- `snAn)nM(6)(I + 6nAn) = Rh(6) + a(6n+l) Tracing back the n + 1 canonical M(6) transformations, = Ao(I + 6Aa ) .
Density = m-hp(z-") = p (m-n_) k Currently, the nth-turn ,th-tllrn the required h_,_rn results are to be reported. 48 50 order and the phase-space prr_tqlp I"nn(_X"J _rfff 1_ rg_........ _d D_f area _ ; 1 _d 5 Dispersed " Betatron Motion As discussed in Chapter 2, there is always energy spread around the nominal energy in the particle beam. The energy spread causes closed-orbit spread. These dispersed closed orbits with respect to the reference orbit are functions of the longitudinal position, s, and of the energy deviation, 6 = AE/Eo (or the off-momentum, 6 =/Xp/po); that is, • = where Xc is a vector representing the transverse phase-space dispersed closed orbit, and its transpose is given by = coordinates of the yo,p ,o).