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The most goal of those lectures is first to in short survey the basic con­ nection among the illustration idea of the symmetric crew Sn and the speculation of symmetric services and moment to teach how combinatorial equipment that come up obviously within the conception of symmetric capabilities result in effective algorithms to specific quite a few prod­ ucts of representations of Sn when it comes to sums of irreducible representations.

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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).

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