Construction of symplectic maps for nonlinear motion of particles in accelerators

Abstract

LBL-34035 07/30/'33We explore an algorithm for construction of symplectic maps to describe nonlinear particle motion in circular accelerators. We emphasize maps for motion over one or a few full turns, which may provide an economical way of studying

Numerical Construction of the Poincaré Map, with Application to Accelerators

“…(Among all Fourier modes with |m i | ≤ M, a great many are found to be negligible, even with a mode cutoff as small as M = 8.) (2) We do the interpolation in I with a Kronecker-product B-spline basis. (This saves a lot of time, since at any point only a few of the basis functions are nonzero.)…”

“…We seek an approximation T 1 ≈ T 0 in a region U of phase space. To meet the symplectic condition we construct the mixed-variable, canonical generator G of T 1 , which defines T 1 implicitly [1,2,3]. Attempts at explicit representations of T 1 have shown either excessive violation of symplecticity or difficulties in controlling accuracy; see Ref.…”

“…Attempts at explicit representations of T 1 have shown either excessive violation of symplecticity or difficulties in controlling accuracy; see Ref. [2] for a survey.…”

Full-turn symplectic map from a generator in a Fourier-spline basis

“…For the m = 0 mode, one must be careful to consider the fact that the derivatives of the basis functions are linearly dependent. Details of this can be found in [4]. It is advantageous to choose B-splines for the basis functions.…”

“…This paper is a brief summary of our mapping method. Details and associated references can be found in [4].…”