ABSTRACT
After introducing the reference trajectory and the elements, such as bending magnets, that define it, a Matlab program to visualize the trajectory in space is introduced and used to produce two-dimensional plots and three-dimensional solid models of the accelerator. In a next step, using a Hamiltonian approach, theforces, which are conventionally defined in the laboratory system, are transformed into a coordinate system that moves with a particle on the reference trajectory. In the same step, the particle positions and momenta are transformed into the phase-space coordinates that are commonly used to describe the motion of particles with respect to a reference particle. First, the coordinates of single particles are expressed in the co-moving system, followed by the description of ensembles of many particles–the bunch–by multi-variate distribution functions. Then, the moments of the distributions are introduced as a more efficient description of the ensemble, which leads to the commonly used characterization of a bunch in terms of its centroid position and the beam- or sigma-matrix.
