Solvable map representation of a nonlinear symplectic map

Abstract

The evolution of a particle under the action of a beam transport system can be represented by a nonlinear symplectic map M. This map can be factorized into a product of Lie transformations. The evaluation of any given lie transformation in general requires the summation of an infinite number of terms. There are several ways of dealing with this difficulty: The summation can be truncated, thus producing a map that is nonsymplectic, but still useful for short term tracking. Alternatively, for long term tracking, the Lie transformation can be replaced by some symplectic map that agrees with it to some order and can be evaluated exactly. This paper shows how this may be done using solvable symplectic maps. A solvable map gives rise to a power series that either terminates or can be summed explicitly. This method appears to work quite well in the various examples that we have considered.