Distinction between difference and differential equations of motion for synchrotron motion

Abstract

The meticulous development of the differential equations of motion (EOM) and related analysis for synchrotron oscillation has extended over decades [see, for example, references [1], [2], and [3]], but the parallel treatment of the difference equations (turn-by-turn mapping) is less developed. The author has written the difference equations for the mapping[4], that is to say the difference equation EOM, but has not carried the fundamental development much further. A companion paper[5] is an effort to reconcile difference EOM and differential EOM. Practically one finds that both techniques generally give the same numerical solution when applied to precisely the same physical system. However, it is incorrect to assume that the two methods are mathematically equivalent; there are circumstances in which the distinction between them is material. This note develops the commonality of the two formalisms.