Lie Algebraic Methods for Particle Accelerator Theory.

The problem of determining charged particle behavior in electromagnetic fields falls within the realm of Hamiltonian dynamics. Consequently, the motion of a charged particle in an accelerator is amenable to description using a variety of the mathematical structures inherent to a Hamiltonian system. Amongst the most useful of these are a hierarchy of Lie algebras and Lie groups defined via the Poisson bracket. In this thesis we make new applications of several concepts from the theory of Lie groups and Lie algebras to certain types of calculations encountered in accelerator science. We introduce a variety of techniques from the theory of Lie algebras which prove useful in developing a description of charged particle motion. Applications of these techniques are then made. A preponderence of this thesis concerns itself with computation of particle trajectories using Lie algebraic methods. An analytical perturbation method for computing particle trajectories is developed and application made to a variety of beam -line elements common in accelerators. In addition, methods for numerical computations based on a Lie algebraic formalism are introduced. An algebraically based tracking code (MARYLIE) is presented as an example of the economy of calculation made possible through use of Lie algebraic methods. This code is designed to perform ray traces through beam lines (comprised of any of a variety of common elements) accurately through nonlinear terms of third order in deviations from beam-line design values. Comparison is made with current matrix theories (which generally include only second order nonlinearities). The use of Lie algebraic methods in chromaticity calculations is discussed. A possible connection between the generators of a Lie transformation of a beamline and the chromaticity of that beamline is presented. Possible problems of future interest are briefly outlined and suggestions for further applications of Lie algebraic methods made.