High order symplectic integrators for perturbed Hamiltonian systems

  • Jacques Laskar
  • Philippe Robutel


A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB was given in (McLachlan, 1995). We give here a constructive proof that for all integer p, such integrator exists, with only positive steps, and with a remainder of order Opε + τ2ε2), where τ is the stepsize of the integrator. Moreover, we compute the analytical expressions of the leading terms of the remainders at all orders. We show also that for a large class of systems, a corrector step can be performed such that the remainder becomes Opε +τ4ε2). The performances of these integrators are compared for the simple pendulum and the planetary three-body problem of Sun–Jupiter–Saturn.

symplectic integrators Hamiltonian systems planetary motion Lie algebra 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abramovitz, M. and Stegun, I.: 1965, Handbook of Mathematical Functions, Dover, New York.Google Scholar
  2. Bourbaki, N: 1972, Groupes et algèbres de Lie, Eléments de Mathématiques, Hermann, Paris.Google Scholar
  3. Chambers, J. and Murison, M. A.: 2000, ‘Pseudo-high-order symplectic integrators’, Astron. J. 119, 425–433.Google Scholar
  4. Channell, P. J. and Neri, F. R.: 1996, ‘An introduction to symplectic integrators’, Integration algorithms and classical mechanics (Toronto, ON, 1993), Fields Inst. Commun. 10, Amer.Math. Soc., Providence, RI, 45–58.Google Scholar
  5. Duncan, M., Levison, H. and Lee, M. H.: 1998, ‘A multiple time step symplectic algorithm for integrating close encounters’, Astron. J. 116, 2067–2077.Google Scholar
  6. Forest, E. and Ruth, R. D.: 1990, ‘Fourth-order symplectic integration’, Phys. D 43(1), 105–117.Google Scholar
  7. Forest, E.: 1992, ‘Sixth-order Lie group integrators’, J. Comput. Phys. 99(2), 209–213.Google Scholar
  8. Forest, E.: 1998, Beam Dynamics. A New Attitude and Framework, Harwood Academic Publishers.Google Scholar
  9. Koseleff, P.-V.: 1993, ‘Relations among Lie formal series and construction of symplectic integrators’, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, (San Juan, P. R. 1993), Lecture Notes in Comput. Sci., 673, Springer, Berlin, pp. 213–230.Google Scholar
  10. Koseleff, P.-V.: 1996, ‘Exhaustive search of symplectic integrators using computer algebra’, in Integration algorithms and classical mechanics (Toronto, ON, 1993), Fields Inst. Commun. 10, Amer. Math. Soc., Providence, RI, 103–119.Google Scholar
  11. Laskar, J.: 1990, ‘Systèmes de variables et èlèments’, in: D. Benest and C. Froeschlè (eds) Les Mèthodes Modernes de la Mècanique Cèleste, (Goutelas, 1989), Editions Frontiéres.Google Scholar
  12. Laskar, J. and Robutel, P.: 1995, ‘Stability of the planetary three-body problem. I Expansion of the Planetrary Hamiltonian’, Celest. Mech. & Dyn. Astr. 62, 193–217.Google Scholar
  13. McLachlan, R. I.: 1995, ‘Composition methods in the presence of small parameters’, BIT 35(2), 258–268.Google Scholar
  14. McLachlan, R. I., Quispel, G. R. W. and Turner, G. S.: 1998, ‘Numerical integrators that preserve symmetries and reversing symmetries’, SIAM J. Numer. Anal. 35(2), 586–599Google Scholar
  15. Neri, F.: 1988, Lie Algebras and Canonical Integration, Dept. of Physics, University of Maryland, preprint.Google Scholar
  16. Ruth, R.: 1983, ‘A canonical integration technique’, IEEE Tran. Nucl. Sci. 30, 2669–2671.Google Scholar
  17. Touma, J. and Wisdom, J.: 1994, ‘Lie-Poisson integrators for rigid body dynamics in the Solar System’, Astron. J. 107, 1189–1202.Google Scholar
  18. Saha, P. and Tremaine, S.: 1992, ‘Symplectic integrators for Solar System dynamics’, Astron. J. 104, 1633–1640.Google Scholar
  19. Suzuki, M.: 1991, ‘General theory of fractal path integrals with applications to many-body theories and statistical physics’, J. Math. Phys. 32(2), 400–407.Google Scholar
  20. Suzuki, M.: 1992, ‘General theory of higher-order decomposition of exponential operators and symplectic integrators’, Phys. Lett. A 165(5-6), 387–395.Google Scholar
  21. Wisdom, J. and Holman, M.: 1991, ‘Symplectic Maps for the N-Body Problem’, Astron. J. 102(4), 1528–1538.Google Scholar
  22. Wisdom, J., Holman, M. and Touma, J.: 1996, ‘Symplectic correctors’, Integration algorithms and classical mechanics (Toronto, ON, 1993), Fields Inst. Commun., 10, Amer. Math. Soc., Providence, RI, 217–244.Google Scholar
  23. Yoshida, H.: 1990, ‘Construction of higher order symplectic integrators’, Phys. Lett. A 150(5-7), 262–268.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Jacques Laskar
  • Philippe Robutel

There are no affiliations available

Personalised recommendations