A Modular Approach for Beam Lines Design

  • Serge N. Andrianov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6885)


We discuss advantages of numerical simulation based on symbolic presentations of beam line dynamical models. In some previous papers, some of these features were discussed. In this paper, we demonstrate how the symbolic presentation of necessary information can provide an in-depth study of different features of complex systems. For this purpose, we suggest a modular principle for all levels of the modeling and optimization procedures. This principle is based on so-called LEGO objects, which have both symbolic and numerical representation. For beam line design, it is necessary to support three types of similar objects. The first of them contains all necessary objects for beam line components description, the second contains all objects which correspond to particle beam models, and the third contains all objects corresponding to a transfer map (“a beam propagator”). In the suggested approach, the beam propagator is presented as a set of two-dimensional matrices describing different kinds of beam or beam line properties up to some approximation order. These matrices can be computed both in symbolic and numerical forms up to the necessary approximation order of the nonlinear effects. An example of practical application is demonstrated.


Symbolic algebra LEGO object Lie algebraic methods beam physics 


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  1. 1.
    Cai, Y., Donald, M., Irwin, J., Yan, J.: LEGO: A Modular Accelerator Design Code. SLAC-PUB-7642 (August 1997)Google Scholar
  2. 2.
    Dragt, A.J.: Lie Methods for Nonlinear Dynamics with Applications to Accelerator Physics, p. 1805. University of Maryland, College Park (2011), Google Scholar
  3. 3.
    Dragt, A.J.: Lectures on nonlinear orbit dynamics. In: AIP Conf. Proc., vol. (87), pp. 147–313 (1987)Google Scholar
  4. 4.
    Dragt, A.J.: Lie Algebraic Treatment of Linear and Nonlinear Beam Dynamics. In: Annual Review of Nuclear and Particle Science, vol. 38, pp. 455–496 (1988)Google Scholar
  5. 5.
    Andrianov, S.N.: The explicit form for Lie transformations. In: Proc. Fifth European Particle Accelerator Conference EPAC 1996, SITGES (Barcelona, Spain), pp. 998–1000. Barselona (1996)Google Scholar
  6. 6.
    Andrianov, S.N.: Matrix representation of the Lie algebraic methods for design of nonlinear beam lines. In: AIP Conf. Proc., N.Y, vol. (391), pp. 355–360 (1997)Google Scholar
  7. 7.
    Andrianov, S.N.: Symbolic computation of approximate symmetries for ordinary differential equations. Mathematics and Computers in Simulation 57(3-5), 147–154 (2001)Google Scholar
  8. 8.
    Andrianov, S.N.: Lego-Technology Approach for Beam Line Design. In: Proc. EPAC 2002, Paris, France, pp. 1667–1669 (2002)Google Scholar
  9. 9.
    Andrianov, S.N.: Dynamical Modeling of Control Systems for Particle Beams. SPbSU, Saint Petersburg (2004) (in Russian)Google Scholar
  10. 10.
    Andrianov, S.N.: A role of symbolic computations in beam physics. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2010. LNCS, vol. 6244, pp. 19–30. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Dragt, A.J., Finn, J.M.: Lie series and invariant functions for analytic symplectic maps. J. Math. Phys. 17(12), 2215–2227 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sanz-Serna, J.M.: Symplectic integrators for Hamiltonian problems: an overview. Acta Numerica 1, 243–286 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ruth, R.D.: A canonical integration technique. IEEE Trans. Nucl. Sci. 30, 2669 (1983)CrossRefGoogle Scholar
  14. 14.
    Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262 (1990)Google Scholar
  15. 15.
    Forest, E.: Canonical integrators as tracking codes. In: AIP Conf. Proc., vol. 184, pp. 1106–1136. American Institute of Physics, New York (1989)CrossRefGoogle Scholar
  16. 16.
    Andrianov, S., Edamenko, N., Podzivalov, E.: Some problems of global optimization for beam lines. In: Proc. PHYSCON 2009, Catania, Italy, September 1-4 (2009), Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Serge N. Andrianov
    • 1
  1. 1.Faculty of Applied Mathematics and Control ProcessesSaint Petersburg State UniversitySaint PetersburgRussian Federation

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