A book on Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics is currently in preparation. The most recent version, 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 60 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 6 November 2009. This version
has some 1,668 pages (210 pages more than the 24 January 2009 version). All known errors
have been corrected, and the following additions have been made:
a) A brief discussion of Symbolic Dynamics has been added to an Exercise in Chapter
1. Section 1.4.3, Stroboscopic Maps and Duffing Equation Example, is complete, and an
associated Appendix J (Feigenbaum Cascade Denied) has been added. b) Chapter 3 on Symplectic Matrices now contains a proof of the Krein-Moser theorem,
and an associated Appendix O on Quadratic Forms has also been added. A Section 3.12
has been added on General Symplectic Forms, Darboux Transformationas, and Variant
Symplectic Groups. c) Exercises have been added to Section 6.2 of Chapter 6 on Symplectic Maps to show
that Galilean, Lorentz, Poincare, and gauge transformations are all symplectic maps. It
is also shown that the Galilean group is a contraction of the Poincare group in the
limit that the speed of light c is taken to be infinite. d) An Exercise has been added to Section 7.2 of Chapter 7 on Lie Transformations and
Symplectic Maps to illustrate how Lorentz (and hence Poincare) transformations can be
written in terms of Lie transformations. e) Work has begun on Chapter 11, Transfer Maps for Idealized Straight Beam-line
Elements. f) Chapter 13 on Cylindrical Harmonic and Taylor Expansions is complete. g) Chapter 14 on Realistic Transfer Maps for Straight Beam-Line Elements is complete.
Chapters 13 through 17, and their planned sequel, Chapter 19 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 14.1.1. In
this approach all fringe-field and higher-order multipole effects are automatically
included. h) Chapter 15 on Tools for Numerical Implementation is complete. i) The sections of Chapter 16 (Numerical Benchmarks) and Chapter 17 (Smoothing and
Insensitivity to Errors) that deal with the use of circular and elliptic cylinders are
complete. The sections dealing with rectangular cylinders are yet to be written. The
Numerical Benchmarks chapter illustrates that accuracies of a few parts in 10,000 or
better can be achieved for maps through 7th order, and the Smoothing and Insensitivity
to Errors chapter illustrates that the use of surface methods mitigates the problem of
numerical noise in the computation of high derivatives. j) Work has begun on Chapter 18, Applications of Cylindrical Surface Methods. Section
18.2 on soleniods is now complete. k) Work has begun on Chapter 19, Realistic Transfer Maps for Curved Beam-line
Elements. l) Apart from some polishing, Sections 22.12 and 22.13 of Chapter 22
(General Maps) are now complete. These Sections illustrate the remarkable fact that a
Taylor approximation to the stroboscopic Duffing map, including parameter dependence,
can reproduce the full period doubling Feigenbaum cascade and associated strange
attractor of the exact map. m) Work has begun on Chapter 25, Solved and Unsolved Polynomial Orbit Problems:
Invariant Theory. Substantial text has been written including an Exercise illustrating
Krein collisions. n) Work has begun on Chapter 27, Optimal Evaluation of symplectic Maps.