  To a very good approximation, particularly for hadron machines,
charged-particle trajectories in accelerators obey Hamiltonian mechanics.
During routine storage times of eight hours or more, such particles execute
some $10^{8}$ revolutions about the machine, $10^{10}$ oscillations about the
design orbit, and $10^{13}$ passages through various bending and focusing
elements. Prior to building, or modifying, such a machine, we seek to identify
accurately the long-term behavior and stability of particle orbits over such
large numbers of interactions. This demanding computational effort does not
yield easily to traditional methods of symplectic numerical integration,
including both explicit Yoshida-type and implicit Runge-Kutta or Gaussian
methods. As an alternative, one may compute an approximate one-turn map and
then iterate that map. We describe some of the essential considerations and
techniques for constructing such maps to high order and for realistic magnetic
field models. Particular attention is given to preserving the symplectic
condition characteristic of Hamiltonian mechanics.
