ABSTRACT
In this chapter, the authors review through the gutter example some elements of Lagrangian and Hamiltonian dynamics. Their ultimate goal is to write a Hamiltonian for a section of gutter which uses distance along the gutter as the independent variable. The authors show that the averaged Hamiltonian of the straight gutter contains many terms. They provide without proof an ultra relativistic Hamiltonian for large machines. The authors describe the techniques needed to approximate or compute the motion through a segment of gutter, and discuss how the various segments are put together. They point out that automatic differentiation is not a "small difference" numerical procedure. It is an exact method. The extraction of maps by automatic differentiation is based on the algebra of truncated power series which incidently is not a differential algebra even though it is isomorphismic to one.
