Relations among Lie formal series and construction of symplectic integrators

  • P. -V. Koseleff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 673)

Abstract

Symplectic integrators are numerical integration schemes for hamiltonian systems. The integration step is an explicit symplectic map. We find symplectic integrators using universal exponential identities or relations among formal Lie series. We give here general methods to compute such identities in a free Lie algebra. We recover by these methods all the previously known symplectic integrators and some new ones. We list all possible solutions for integrators of low order.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    Bourbaki, N.: Groupes et algèbres de Lie, Éléments de Mathématiques, Hermann, Paris, 1972.Google Scholar
  2. [2]
    Cary, J.R.: Lie Transform Perturbation Theory for Hamiltonian Systems, in Physics Reports, North-Holland Publishing Company 79-2 (1981), 129–159.Google Scholar
  3. [3]
    Deprit, A.: Canonical transformations depending on a small parameter, Cel. Mech. 1 (1969), 12–30.Google Scholar
  4. [4]
    Dragt, A. J., Finn, J. M.: Lie Series and invariant functions for analytic symplectic maps, J. Math. Physic 17 (1976), 2215–2227.Google Scholar
  5. [5]
    Dragt, A. J., Healy, L. M.: Concatenation of Lie Algebraic Maps, in Lie Methods in Optics II, Lec. Notes in Physics 352 (1988).Google Scholar
  6. [6]
    Finn, J. M.: Lie Series: a Perspective, Local and Global Methods of nonlinear Dynamics, Lec Notes in Physics 252 (1984), 63–86.Google Scholar
  7. [7]
    Forest, E., Ruth, D.: Fourth-Order Symplectic Integration, Physica D 43 (1990), 105–117.Google Scholar
  8. [8]
    Koseleff, P.-V., Thèse de troisième cycle, École Polytechnique, (to appear).Google Scholar
  9. [9]
    Michel, J.: Bases des Algèbres de Lie Libres, Etude des coefficients de la formule de Campbell-Hausdorff, Thèse, Orsay, 1974.Google Scholar
  10. [10]
    Perrin, D.: Factorization of free monoids, in Lothaire M., Combinatorics On Words, Chap. 5, Addison-Wesley (1983).Google Scholar
  11. [11]
    Petitot, M.: Algèbre non commutative en Scratchpad: application au problème de la réalisation minimale analytique, Thèse, Université de Lille I, 1991.Google Scholar
  12. [12]
    Steinberg, S.: Lie Series, Lie Transformations, and their Applications, in Lie Methods in Optics, Lec. Notes in Physics 250 (1985).Google Scholar
  13. [13]
    Suzuki M.: Fractal Decomposition of Exponential Operators with Applications to Many-Body Theories and Monte Carlo Simulations, Ph. Letters A 146 (1990), 319–323.Google Scholar
  14. [14]
    Wisdom J., Holman, M.: Symplectic Maps for the N-Body Problem, The Astr. J. 102(4) (1991), 1528–1538.Google Scholar
  15. [15]
    Yoshida, H.: Conserved Quantities of Symplectic Integrators for Hamiltonian Systems, Physica D (1990).Google Scholar
  16. [16]
    Yoshida, H.: Construction Of Higher Order Symplectic Integrators, Ph. Letters A 150, (1990), 262–268.Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • P. -V. Koseleff
    • 1
  1. 1.Aleph et Géode Centre de Mathématiques, École PolytechniquePalaiseau

Personalised recommendations