Abstract
We describe a method and a way of thinking which is ideally suited for the study of systems represented by canonical integrators. Starting with the continuous description provided by the Hamiltonian, we replace it by a succession of preferably canonical maps. The power series representation of these maps can be extracted with a computer implementation of the tools of non-standard analysis and analyzed by the same tools. For a nearly integrable system, we can define a Floquet ring in a way consistent with our needs. Using the finite time maps, the Floquet ring is defined only at the locations s; where one perturbs or observes the phase space. At most the total number of locations is equal to the total number of steps of our integrator. We can also produce pseudo-Hamiltonians which describe the motion induced by these maps.
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References
A.J. Dragt and E. Forest, J. Math. Phys. 24, 2734 (1983).
The derivation of this result uses a differential equation for the symplectic map M and the concept of the intermediate (or Dirac) representation. It is explained in detail in reference 1. One can consult A. Messiah Mecanique Quantique, (Dunod 1965) p. 270 for a familiar quantum mechanical version of the derivation.
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A nonlinear map expanded into a power series can be factorized into a linear and a purely nonlinear part. One can either compute a single Lie operator for the nonlinear part or factor it into a product of Lie transformations of increasing degree. See reference 1 or the original derivation in: A.J. Dragt and J.M. Finn, J. Math. Phys. 20, 2649 (1979).
A description of the FORTRAN implementation of differential algebra by M. Berz is to be found in SSC-152 or SSC-166 technical reports of the Superconducting Super Collider Central Design Group. Both reports have been accepted for publication in Particle Accelerators.
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See reference 5 (SSC-166) for a detailed description of the differential algebra implementation of the normal form process.
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Independently of our Hamiltonian-free approach, it has been developed and used by another group. See: A. Bazzani, P. Mazzanti, G. Servizi, and G. Thrchetti, “Normal Forms for Hamiltonian Maps and Non Linear Effects in a LHC Model”, CERN SPS/88-2 (AMS) LHC note 66 (European Organisation for Nuclear Research, 1988).
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Forest, E., Berz, M. (1989). Canonical integration and analysis of periodic maps using non-standard analysis and Lie methods. In: Wolf, K.B. (eds) Lie Methods in Optics II. Lecture Notes in Physics, vol 352. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0012744
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DOI: https://doi.org/10.1007/BFb0012744
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