ABSTRACT
In this chapter, the author discusses symplectic integration. The Spanish school of Sanz-Serna and collaborators have worked extensively on explicit and implicit formulation of symplectic Runge-Kutta algorithms. All explicit symplectic schemes involve breaking the Hamiltonian into exactly solvable parts and treating each part on an equal footing. The most important concept is that of "uncontrolled transverse approximations" because the purpose of ring studies is to determine the effect of errors and the new magnets on the transverse stability of the ring. Implicit integrators are usually avoided by accelerator physicists but are often studied in other fields. Alex Dragt was interested to know how the small angle standard kick code could produce the correct chromatic behavior while truncating the Hamiltonian in the leading order in angle.
