Lie methods, exact map computation, and the problem of dispersion in space charge dominated beams

The concept and use of 'transfer maps' has become a fundamental ingredient in a modern description of single-particle beam dynamics. Lie methods, in particular, provide a useful framework for map computation. In the autonomous case calculation of a transfer map is equivalent to evaluating a Lie transformation. In general this is a nontrivial problem. The first part of the Dissertation contains a discussion of two possible ways to compute Lie transformations based on the theory of normal forms for Lie maps and the Scaling, Splitting, and Squaring algorithm. Implementation of the latter in the code MARYLIE5.0 is also discussed.; In the second part we study fringe field effects in magnets and develop suitable methods to describe these effects without idealization of field profiles. In particular, we introduce a method for the exact computation of transfer maps using a realistic description of magnetic fields. The magnetic field data can be obtained either from numerical computation with the aid of a 3D electromagnetic code or from measurements with spinning coils. The method is applied to study beam dynamics in the Large Hadron Collider. In other special situations one can investigate fringe-fields effects using a realistic analytical model. This is the case for the University of Maryland E-Ring, for which we present a thorough study of single particle dynamics. Using the analytical model we show how the third-order intrinsic aberrations, due to fringes, decrease with increasing magnet aperture.; In the third part of the Dissertation we shift our attention to space charge effects in small rings and analyze the interplay between space charge and dispersion. We work out a novel set of equations that can be used to obtain simultaneous matching of rms-envelopes and the dispersion function in space-charge dominated beams. Central to the derivation of the new equations is the existence of a new linear invariant that generalizes the rms emittance to the case where bending magnets and an energy spread are present.