A book on *Lie Methods for
Nonlinear Dynamics with Applications to Accelerator Physics* is currently in
preparation. The most recent version, with the additions described below, is presently available in
draft
form. It is expected that newer versions will become available approximately
every few months. To download either the Table of Contents, or the Full Book, click on one of
the links below. If you are using an older version of Adobe, you may have
trouble. However, the newer versions of Adobe should work fine. Patience may be required
in downloading the Full Book. Its file is large, approximately 65 MB. Some figures in
the book may not be viewable on the screen because they consist of many very small dots; however, they
all should print.

Version of 19 November 2020: This version has some 2,726 pages (1,270 pages more than the 24 January 2009 version). Most Chapters (now 39 in number) and most Appendices (now 25 in number) are essentially complete or at least contain substantial material. Below is a list of recent additions that have been made and work that is in progress:

a) Section 2.3.4 on Runge-Kutta integration has been added to describe the use of Butcher tableaux and the relation between order and the number of stages in the cases of both explicit and implicit methods. First Same As Last (FSAL) methods are also described. Correspondingly, Appendix B has been enlarged to cover the embedded Runge-Kutta methods Fehlberg 4(5) and Dormand-Prince 5(4).

b) Section 2.4, now called "Finite-Difference/Multistep/Multivalue Methods", has been enlarged to place Adams' method in the general context of multistep methods. The additional text defines multistep methods, describes their maximum order, illustrates stability and the existence of parasitic solutions, describes the first Dahlquist barrier, states a convergence theorem, and shows that Adams' method has optimal stability properties.

c) Section 5.13 in Chapter 5 describing how symplectic and symmetric matrices are related by Darboux Moebius transformations has been enlarged. Sections 6.5 through 6.7 in Chapter 6 describing how symplectic maps are related to gradient maps by Darboux Moebius transformations, which provide a general theory of generating functions, have been enlarged.

d) The material on geometric integration, originally in Chapter 10, has been moved and made part of a separate Chapter 11. This new chapter, now devoted to a survey of integration on manifolds, is still under construction.

e) A separate Chapter 12, also under construction, will be devoted to symplectic integration.

f) Chapter 15 has been enlarged to now cover Spherical Harmonic Expansions as well as Cylindrical Harmonic Expansions, including the construction of minimal vector potentials. Related Appendices U and V on spherical polynomial vector fields and reference-plane rotations have been added. The use of minimal vector potentials is key to the proper termination of fringe fields. Chapters 15 through 21 on Realistic Transfer Maps for Straight Beam-Line Elements, and their planned sequels, Chapter 22 through 25 on Realistic Transfer Maps for Curved Beam-Line Elements, describe how, for the first time, it is now possible by using surface methods to compute realistic transfer maps based on 3-dimensional field data on a mesh as provided by various electromagnetic solvers. See Figures 17.1.1 and 22.1.1. In this approach all fringe-field and higher-order multipole effects are automatically included.

g) An Appendix X that treats Lie Methods for Light Optics is under construction.

h) Material has been added in Chapters 6 through 8 on the Lorentz group: its derivation without the initial assumption of linearity, its SL(2.C) covering group and representations, its Sp(2,R) subgroup and Wigner rotations, its relation to the Dirac equation, and the use of Sp(2,R) and orthogonal polar decomposition for concatenating coplanar boosts.