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 (with the exception of the stereographic view of the dynamic aperture of the
Henon map in Section 1.2, which has yet to be made in digital form).

Version of 3 April 2014: This version has some 2,224 pages (766 pages more than the 24 January 2009 version). Most chapters, now 35 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 tableaus 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) 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.

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

e) Chapter 16 on Realistic Transfer Maps for Straight Beam-Line Elements is being enlarged to include a section on terminating fringe fields. A related Appendix U on spherical polynomial vector fields has been added. Chapters 15 through 20, and their planned sequel, Chapter 21 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 Figure 16.1.1. In this approach all fringe-field and higher-order multipole effects are automatically included. Chapter 20 now contains a description of fringe-field fall off for straight beam-line elements. Work continues on Chapter 21.