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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References
N. Bourbaki, Elements of Mathematics, Lie Groups and Lie Algebras (Addison-Wesley, Reading, Massachusetts, 1971); Part I: Chapters 13.
D. R. Douglas, Ph.D. dissertation, University of Maryland Physics Department Technical Report (1982).
A.J. Dragt, Lectures on Nonlinear Orbit Dynamics, in: Physics of High Energy Particle Accelerators, Proceedings of the 1981 Summer School on High Energy Particle Accelerators, New York, R.A. Carrigan, F.R. Huson, and M. Month, eds., American Institute of Physics, Conference Proceedings Number 87, 1982, p. 147.
A.J. Dragt, and J.M. Finn, Lie series and invariant functions for analytic symplectic maps, J. Math. Phys. 17, 2215 (1976).
A.J. Dragt, and E. Forest, Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods, J. Math. Phys. 24, 2734 (1983).
A.J. Dragt et al., Marylie 3.0 manual, University of Maryland Technical Report (1988).
A.J. Dragt, L.M. Healy, F. Neri, R.D. Ryne, D.R. Douglas, and E. Forest, MARYLIE 3.0-A Program for Nonlinear Analysis of Accelerators and Beamline Lattices, IEEE Trans. Nucl. Sci. NS-32, 2311 (1985).
A.J. Dragt, F. Neri, G. Rangarajan, D. Douglas, L.M. Healy, and R.D. Ryne, Lie algebraic treatment of linear and nonlinear beam dynamics, Ann. Rev. Nucl. Part. Sci. 38, 455–496 (1988).
M. Hausner and J. Schwartz, Lie Groups Lie Algebras (Gordon and Breach, New York, 1968).
L.M. Healy, Ph.D. dissertation, University of Maryland Physics Department Technical Report (1986).
L.M. Healy and A.J. Dragt, Lie algebraic methods for treating lattice parameter errors in accelerators. In: Proceedings of the 1987 IEEE Particle Accelerator Conference Vol. 2, (1987), p. 1060 et seq.
L.M. Healy and A.J. Dragt, Computation of nonlinear behavior of inhomogeneous Hamiltonian systems using Lie algebraic methods (in preparation).
SMP Manual, Inference Corporation, 1985.
F.J. Murray and K.S. Miller, Existence Theorems (New York University Press, New York, 1954).
V.S. Varadarajan, Lie Groups, Lie Algebras, and their Representations (Springer-Verlag, New York, 1984).
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© 1989 Springer-Verlag
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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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