Convention and convenience in the beam physics community ask that for a given apparatus we are looking for a map that takes the initial state of particles as they enter the device to that of a final state as they exit. However, in general, we arenít looking to provide a relationship between the initial state of particles and their final states at a given time; rather, we are looking for the relationship between the two at two different planes in space. Namely, we are looking for a map that relates the state of particles zi on an initial plane s0 = 0 to that of the state zf on the final plane sf as they travel through a specified electromagnetic field. This map can be obtained to arbitrary order within the framework of the code COSY INFINITY through the use of differential algebras (DA).
Often in dynamical systems, we ask for a mathematical construct called a Poincaré section in which we keep track of locations on a given plane that an integrated orbit passes through. This generally makes it easier for one of the requirements of performing a Poincaré section in a differential algebraic framework...
[This is an extract from the beginning page of the paper.]
R. Jagasia, M. Berz, B. Loseth, Microscopy and Microanalysis 21 Suppl. 4 (2015) 14
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