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.