ABSTRACT
The ring is a collection of maps necessary for the description/calculation of physical phenomena that interest the scientist. The beauty of a map based theory is that it keeps the number of maps to an absolute minimum, as dictated by the experimental conditions. The ring is a finite and ordered collection of charts with the maps needed to propagate from one chart to the next. When dealing with pure tracking/experimental data, it is quite clear that we "push" from chart to chart the state variables which describe our system. This is experimentally self-evident: we cannot have an infinite number of beam position monitors and diagnostics around a ring. In the rarified air of mathematics and theoretical dynamics, one is interested primarily in the topological nature of phase space. This has been true since Poincaré. In this book, we also emphasize geometrical features because we find this to be more powerful for practical and theoretical applications.
