Abstract
Time evolution in a Hamiltonian system may be represented by a transfer map, which in turn may be represented as a product of Lie transformations factored by order. Two such products in succession may be concatenated into a single product. It is possible to do this even when the Lie transformations include inhomogeneous terms. Rules are given for combining and ordering Lie transformations in the general case through sixth order. These rules are presented in an algorithmic fashion suitable for manipulation by computer.
Such techniques have applications to many Hamiltonian systems, including accelerator beam dynamics and optics. The concatenation could represent, for instance, the combined effects of two successive beamline elements in a particle accelerator. In this case, inhomogeneous terms can arise when there are placement, alignment, or powering errors.
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Healy, L.M., Dragt, A.J. (1989). Concatenation of Lie algebraic maps. In: Wolf, K.B. (eds) Lie Methods in Optics II. Lecture Notes in Physics, vol 352. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0012745
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DOI: https://doi.org/10.1007/BFb0012745
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