ABSTRACT
This chapter deals with the one-turn map and shows that all the dynamics are contained in it. Conceptually and practically, it is the most important object of ring dynamics. The chapter argues that the physical ring, the actual piece of hardware, has a physical existence of its own and is not a mathematical abstraction. It also argues that all of this must be mathematically equivalent to the usual Hamiltonian theory which avoided and will keep avoiding when discussing global dynamical concepts. The reader may still want to see how this comes about. The chapter looks at the maximal ring Ring∞, i.e., the Hamiltonian. More precisely, how does one get all the nice analytical formulas of Hamiltonian theory by taking the limit K tending to infinity? The chapter illustrates the dangerous over-kill and extra-complexity of the Hamiltonian approach.
