Diffusion in Hamiltonian Systems with Applications to Twist Maps and the Two Beam Accelerator.

In this thesis I study two problems involving chaotic motion in Hamiltonian systems. In the first, I study the evolution of a distribution of particles in phase space. The motion of the particles is given by a near -integrable Hamiltonian mapping and the variables used to describe the motion are the action-angle variables of the integrable part of the Hamiltonian. The evolution of the distribution of angles is observed to be much faster than the evolution of the distribution of actions. This separation of time scales allows me to average over the angle variables and model the evolution of the action distribution function as a diffusive process. Using the Fermi map as an example I numerically integrate the Fokker-Planck equation for the action and compare the resulting distribution function with direct solutions of the mapping equations. The second moment of the distribution is compared with the diffusion coefficient measured in the numerical experiments. Both show oscillations similar to those found in the standard map. In addition I numerically find the invariant distribution in the Fermi map. I calculate the size of islands surrounding stable fixed points and show that the dips correspond to these islands. The second problem I study involves the motion of electrons in free electron lasers (FELs). Using a variational principle I derive a self consistent three dimensional model for tapered FELs. I then use this model to study the extent of stochastic motion in a particle accelerator. A high current, low energy (20 MeV) beam drives the FELs, producing 1 cm microwaves. These microwaves are used to accelerate a low current beam to energies of 1 TeV. The periodic nature of the acceleration and deceleration of the low energy beam and the interaction between the transverse electron motion and the longitudinal electron motion can lead to stochastic motion. I exhibit design criteria that ensure that detrapping of the low energy beam by this stochasticity is minimal. Numerical integrations of a one dimensional model for the FELs are presented showing various degrees of detrapping. I also explore the effects of islands produced by resonances between the periodic acceleration and the trapped particle motion. (Abstract shortened with permission of author.).