Notes on lie algebraic analysis of achromats

Normal form technique is a powerful method to analyze the achromat problem. Assume the one cell map M{sub cell} = ARe{sup :h{sub 3}}:{sub e}{sup :h{sub 4}}: A{sup {minus}1}, where h{sub 3},h{sub 4} are the normal forms of the generators of the unit cell map, and A is the nonlinear transformation that brings M{sub cell} into its normal form; then the map of the whole system is M{sub N} = M{sub cell}{sup N} = AR{sup N} A{sup {minus}1} = I, provided that we can set e{sup :h{sub 3}}:, e{sup :h{sub 4}}, and R{sup N} to the identity (or only {delta} dependent) maps. Therefore, the conditions to form an achromat are h{sub 3} and h{sub 4} equal to zero (or {delta} dependent only) and the total linear map is identity. In this report, we will apply these conditions to a FODO array (a simple model system) to make it an achromat. We will start from Hamiltonians and work all the way up to obtain the analytical expressions of the required sextupole and octupole strengths