Dedicated symplectic integrators for rotation motions

  • Jacques LaskarEmail author
  • Timothée Vaillant
Original Article


We propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the coefficients of integrators on the moments of inertia of the integrated body, we can construct symplectic dedicated integrators with fewer stages than in the general case for a splitting in three parts of the Hamiltonian. We perform numerical tests to compare the developed dedicated fourth-order integrators to the existing reference integrators for the water molecule. We also estimate analytically the accuracy of these new integrators for the set of the rigid bodies and conclude that they are more accurate than the existing ones only for very asymmetric bodies.


Rotation Symplectic integrators Rigid body Lie algebra 


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Conflicts of interest

The authors declare that they have no conflict of interest.


  1. Blanes, S., Casas, F., Murua, A.: Splitting and composition methods in the numerical integration of differential equations. Boletin de la Sociedad Espanola de Matematica Aplicada SeMA 45, 89–145 (2008)MathSciNetzbMATHGoogle Scholar
  2. Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal). PhD thesis, Mathematical Institute, University of Innsbruck, Austria, English Translation in Journal of Symbolic Computation, Special Issue on Logic, Mathematics, and Computer Science: Interactions. Vol. 41, Number 3–4, pp. 475–511, 2006 (1965)Google Scholar
  3. Celledoni, E., Fassò, F., Säfström, N., Zanna, A.: The exact computation of the free rigid body motion and its use in splitting methods. SIAM J. Sci. Comput. 30(4), 2084–2112 (2008)MathSciNetCrossRefGoogle Scholar
  4. Dullweber, A., Leimkuhler, B., McLachlan, R.I.: Symplectic splitting methods for rigid body molecular dynamics. J. Chem. Phys. 107(15), 5840–5851 (1997)ADSCrossRefGoogle Scholar
  5. Eisenberg, D., Kauzmann, W.: The Structure and Properties of Water. Clarendon Press, Oxford (1969)Google Scholar
  6. Farrés, A., Laskar, J., Blanes, S., Casas, F., Makazaga, J., Murua, A.: High precision symplectic integrators for the solar system. Celest. Mech. Dyn. Astron. 116, 141–174 (2013)ADSMathSciNetCrossRefGoogle Scholar
  7. Fassò, F.: Comparison of splitting algorithms for the rigid body. J. Comput. Phys. 189(2), 527–538 (2003)ADSMathSciNetCrossRefGoogle Scholar
  8. Hairer, E., Vilmart, G.: Preprocessed discrete Moser–Veselov algorithm for the full dynamics of a rigid body. J. Phys. A Math. Gen. 39(42), 13225–13235 (2006)ADSMathSciNetCrossRefGoogle Scholar
  9. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006)zbMATHGoogle Scholar
  10. Jacobi, C.G.J.: Sur la rotation d’un corps. Journal für die reine und angewandte Mathematik 1850(39), 293–350 (1850)CrossRefGoogle Scholar
  11. Koseleff, P.V.: Calcul formel pour les méthodes de Lie en mécanique hamiltonienne. PhD thesis, École Polytechnique (1993)Google Scholar
  12. Koseleff, P.V.: Exhaustive search of symplectic integrators using computer algebra. Integr. Algorithms Class. Mech. Fields Inst. Commun. 10, 103–119 (1996)MathSciNetzbMATHGoogle Scholar
  13. McLachlan, R.I.: Explicit Lie–Poisson integration and the Euler equations. Phys. Rev. Lett. 71(19), 3043–3046 (1993)ADSMathSciNetCrossRefGoogle Scholar
  14. McLachlan, R.I.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16(1), 151–168 (1995)MathSciNetCrossRefGoogle Scholar
  15. Munthe-Kaas, H., Owren, B.: Computations in a free Lie algebra. Philos. Trans. R. Soc. Lond. Ser. A 357(1754), 957–981 (1999)ADSMathSciNetCrossRefGoogle Scholar
  16. Omelyan, I.P.: Advanced gradientlike methods for rigid-body molecular dynamics. J. Chem. Phys. 127(4), 044102 (2007)ADSCrossRefGoogle Scholar
  17. Reich, S.: Momentum conserving symplectic integrators. Physica D 76(4), 375–383 (1994)ADSMathSciNetCrossRefGoogle Scholar
  18. Sheng, Q.: Solving linear partial differential equations by exponential splitting. IMA J. Numer. Anal. 9(2), 199–212 (1989)MathSciNetCrossRefGoogle Scholar
  19. Skokos, C., Gerlach, E., Bodyfelt, J.D., Papamikos, G., Eggl, S.: High order three part split symplectic integrators: efficient techniques for the long time simulation of the disordered discrete nonlinear Schrödinger equation. Phys. Lett. A 378, 1809–1815 (2014)ADSMathSciNetCrossRefGoogle Scholar
  20. Sofroniou, M., Spaletta, G.: Derivation of symmetric composition constants for symmetric integrators. Optim. Methods Softw. 20(4–5), 597–613 (2005)MathSciNetCrossRefGoogle Scholar
  21. Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146, 319–323 (1990)ADSMathSciNetCrossRefGoogle Scholar
  22. Suzuki, M.: General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys. 32, 400–407 (1991)ADSMathSciNetCrossRefGoogle Scholar
  23. Tang, Y.F.: A note on the construction of symplectic schemes for splitable Hamiltonian. J. Comput. Math. 20(1), 89–96 (2002)MathSciNetzbMATHGoogle Scholar
  24. Touma, J., Wisdom, J.: Lie–Poisson integrators for rigid body dynamics in the solar system. Astron. J. 107, 1189–1202 (1994)ADSCrossRefGoogle Scholar
  25. Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150(5–7), 262–268 (1990)ADSMathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.ASD, IMCCE-CNRS UMR8028, Observatoire de Paris, PSL Université, Sorbonne UniversitéParisFrance

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