Lie algebraic treatment of space charge

A major concern in the design and evaluation of charged particle beam transport systems is the accurate computation of single particle trajectories. Calculational tools based on Lie algebraic methods have been extensively developed for this purpose. In many instances these methods are significantly faster than numerical integration but have comparable accuracy. Further, computer programs based on Lie algebraic methods require far less storage than conventional programs, making these techniques the obvious choice for high order calculations.;Lie algebraic methods were originally developed to compute the trajectories of charged particles in externally applied electromagnetic fields. However, during the past several years there has been increased emphasis on the design of beam transport systems that transport high currents. In this case, space charge effects become important. That is, one must compute single particle trajectories by taking into account both the externally applied electromagnetic fields and the (self-consistent) self fields of the particle beam.;The purpose of this thesis is to present a Lie algebraic treatment of space charge. Specifically, we will present a method for finding approximate solutions of the Vlasov-Poisson equations, treated as an initial value problem. We will show that the solutions depend on finding certain symplectic mappings, and we will discuss how to compute these maps given a beam transport system and an initial particle distribution function.