Elementary approximation of exponentials of Lie polynomials

  • Frédéric Jean
  • Pierre-Vincent Koseleff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1255)


Let L=l(x1,..., xm) be a graded Lie algebra generated by x1,..., x m . In this paper, we show that for any element P in L and any order k, exp(P) may be approximated at the order k by a finite product of elementary factors exp(λi,xi,). We give an explicit construction that avoids any calculation in the Lie algebra.


Symplectic Integrator Motion Planning Problem Symmetric Composition Universal Identity Nonholonomie Motion Planning 
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  1. [1]
    Bourbaki, N., Groupes et algèbres de Lie, Éléments de Mathématiques, Hermann, Paris, 1972Google Scholar
  2. [2]
    Falbel, E., Koseleff, P.-V., Parallelograms, Preprint (1996)Google Scholar
  3. [3]
    Jacob, G., Motion Planning by piecewise constant or polynomial inputs, Proceedings of the IFAC Nonlinear Control Systems Design Symposium (1992)Google Scholar
  4. [4]
    Koseleff, P.-V., Relations among Formal Lie Series and Construction of Symplectic Integrators, AAECC'10 proceedings, Lect. Not. Comp. Sci. 673 (1993)Google Scholar
  5. [5]
    Koseleff, P.-V., Exhaustive Search of Symplectic Integrators Using Computer Algebra, Fields Institute Communications 10 (1996)Google Scholar
  6. [6]
    Lafferriere, G., Sussmann H., Motion Planning for controllable systems without drift, Proceedings of the 1991 IEEE International Conference on Robotics and Automation (1991)Google Scholar
  7. [7]
    Laumond, J.P., Nonholonomic Motion Planning via Optimal Control, Algorithmic Foundations of Robotics (1995)Google Scholar
  8. [8]
    MacLachlan, R. I., On the numerical integration of ordinary differential equations by symmetric composition methods, SIAM J. Sci. Comp. 16(1) (1995), 151–168Google Scholar
  9. [9]
    Magnus et al., Combinatorial Group Theory: Presentation of Groups in Terms of Generators and Relations, J. Wiley & Sons, 1966Google Scholar
  10. [10]
    Reutenauer, C., Free Lie algebras, Oxford Science Publications, 1993Google Scholar
  11. [11]
    Steinberg, S., Lie Series, Lie Transformations, and their Applications, in Lie Methods in Optics, Lec. Notes in Physics 250 (1985)Google Scholar
  12. [12]
    Suzuki, M., General Theory of higher-order decomposition of exponential operators and symplectic integrators, Physics Letters A 165 (1992), 387–395Google Scholar
  13. [13]
    Suzuki, M., General nonsymetric higher-order decompositions of exponential operators and symplectic integrators, Physic Letters A 165 (1993), 387–395Google Scholar
  14. [14]
    Yoshida, H., Construction of Higher Order Symplectic Integrators, Ph. Letters A 150, (1990), 262–268Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Frédéric Jean
    • 1
  • Pierre-Vincent Koseleff
    • 1
  1. 1.Équipe Analyse Algébrique, Institut de MathématiquesUniversité Paris 6Paris Cedex 05

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