APPENDIX components of A in the deviation variables x and y . In the straight-element case these Taylor expansions can be found using the circular cylinder harmonic based expan- sion (12). Suppose, for example, we wish to retain in the expansion of the Hamiltonian H appearing in (64) homo- geneous polynomials through degree 8. That is what is required to compute transfer maps through 7 th order. If the design orbit lies on the z axis, as will be the case for any straight multipole such as a solenoid, quadrupole, sextupole, octupole, etc., this expansion is straightfor- ward because in this case the Cartesian coordinates x,y are already deviation variables. We see from (65) that we must retain homogeneous polynomials in the vari- ables x,y through degree 7 in the expansions of A$_{x}$ and A$_{y}$ , and homogeneous polynomials in the variables x,y through degree 8 in the expansion of A$_{z}$ . Inspection of (12) shows that for the cases m = 0 or m odd we then need the C [ n ] $_{m,α}$( z ) with ( m + n ) ≤ 7. For the cases of even m we need the C [ n ] $_{m,α}$( z ) with ( m + n ) ≤ 8. MaryLie/IMPACT is a hybrid code that utilizes Lie-algebraic methods for computing and manipulating charged-particle transfer maps through 5 th order, while space-charge effects are treated using Particle-in-Cell methods [11]. We made use of its Lie algebraic capa- bilities. In the Lie algebraic approach, maps are com- puted and manipulated in Lie algebraic form. Each map describes the transformation of the full six-dimensional phase-space coordinates of a particle as it passes through a given beam-line element. Because of the symplectic na- ture of Hamiltonian motion, through aberrations of order ( n − 1) such a map has the Lie representation where R$_{2}$ describes the linear part of the map, and each f$_{j}$ is a homogeneous polynomial of degree j describing the nonlinear part of the map [4, 12]. The linear map R$_{2}$ and the Lie generators f$_{j}$ are de- termined by solving the equation of motion where is the charged-particle Hamiltonian expressed in terms of deviation variables about the design orbit and expanded in a homogeneous polynomial series. The deviation vari- able Hamiltonian H is determined in turn by the Hamil- tonian K for which path length is the independent vari- able. In Cartesian coordinates and with z taken as the independent variable, K is given by the relation In the case of a wiggler, the design orbit oscillates around the z axis. See Fig. 25. Now the requirement that the Hamiltonian H appearing in (64) be an expan- sion in deviation variables about the design orbit is more involved: The components of A must be expanded about the design orbit. As they stand in (12), they are ex- panded about the z axis. If we are to retain terms in H through degree 8 when a re-expansion is made about the design orbit, then coefficients C [ n ] m,α beyond those listed in the previous paragraph must in principle be included in the calculation. We say that higher-degree terms pro- duce lower-degree terms due to feed down . However, the effect of feed down is small if the oscillations are of small amplitude, as they are for the proposed ILC wiggler. In this example we have found that the approximation of re- taining only the terms in A$_{x}$ , A$_{y}$ , and A$_{z}$ through degree 8 produces changes in the design orbit and the transfer map about the design orbit that are comparable to or smaller than the errors found in the benchmark studies of Section IV. We also remark that this feed-down prob- lem does not arise when the general geometry methods to be described in Part II are employed. Here Φ and A are the electric scalar and magnetic vector potentials, respectively. We conclude that (in the case of no electric fields, Φ = 0) what is needed are Taylor expansions for the M = R$_{2}$ exp(: f$_{3}$ :) exp(: f$_{4}$ :) · · · exp(: f$_{n}$ :) (62) ˙ M = M : − H : (63) H = H$_{2}$ + H$_{3}$ + H$_{4}$ + · · · (64) 18 [9] N. Phinney, N. Toge, and N. Walker, eds., International Linear Collider Reference Design Report, Volume 3: Ac- celerator (International Committe for Future Accelera- tors, 2007), http://www.linearcollider.org/ . [8] N. W. McLachlan, Theory and Application of Mathieu Functions (Dover Publications, Inc., 1964). [7] E. Weisstein, Mathworld - a Wolfram web resource , http://mathworld.wolfram.com/ EllipticCylindricalCoordinates.html . [6] P. Morse and H. Feshbach, Methods of Theoretical Physics , vol. 1 & 2 (McGraw-Hill Book Company, Inc., 1953). [5] M. Venturini and A. Dragt, Nucl. Inst. and Meth. A 427 , 387 (1999). [4] A. J. Dragt, Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics (University of Mary- land, 2009), http://www.physics.umd.edu/dsat . [3] C. E. Mitchell, Ph.D. thesis, University of Maryland, Col- lege Park (2007), http://www.physics.umd.edu/dsat . [2] D. Abell, Phys. Rev. ST Accel. Beams 9 , 052001 (2006). [1] F. B. Hildebrand, Introduction to Numerical Analysis (Dover Books, 1987), 2nd ed., section 3.3, p. 85. He writes “In particular, numerical differentiation should be avoided wherever possible · · · ”. K = − [( p$_{t}$ + q Φ) $^{2}$/c2 − m $^{2}$c2 − ( p$_{x}$ − qA$_{x}$ ) 2 − ( p$_{y}$ − qA$_{y}$ ) $^{2}$]1 / 2 − qA$_{z}$. (65)