Forward Symplectic Integrators for Solving Gravitational Few-Body Problems

  • Siu A. Chin
  • C. R. Chen


We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in addition to the force. We demonstrate the efficiency of these Forward Symplectic Integrators by solving the circular restricted three-body problem in the space-fixed frame where the force on the third body is explicitly time-dependent. These algorithms can achieve accuracy of Runge–Kutta, conventional negative time step symplectic and corrector symplectic algorithms at step sizes five to ten times as large.


symplectic integrators positive time steps three-body orbits long time simulations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Albrecht, J. 1955 An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA, 1999, p. 603Google Scholar
  2. Auer, J., Krotscheck, E., Chin, S. A. 2001‘A fourth-order real-space algorithm for solving local Schrodinger equations’.J Chem Phys11568416846CrossRefGoogle Scholar
  3. Battin, R. H.: 1999, An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA.Google Scholar
  4. Blanes, S., Casas, F., Ros, J. 1999‘Symplectic integrators with processing: A general study’.Siam. J. Sci. Comput21711727CrossRefGoogle Scholar
  5. Chambers, J., Murison, M. A. 2000‘Pseudo-high-order symplectic integrators’Astron. J119425433CrossRefGoogle Scholar
  6. Campostrini, M., Rossi, P. 1990‘A comparison of numerical algorithms for dynamical fermions’Nucl. Phys. B329753764CrossRefGoogle Scholar
  7. Candy, J., Rozmus, W. 1991‘A symplectic integration algorithm for separable Hamiltonian functions’J. Comp. Phys.92230256CrossRefGoogle Scholar
  8. Channell P.J., 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, pp. 45–58Google Scholar
  9. Chin, S.A. 1997‘Symplectic integrators from composite operator factorizations’Phys. Lett. A226344348CrossRefGoogle Scholar
  10. Chin, S.A., Kidwell, D.W. 2000‘Higher-order force gradient symplectic algorithms’Phys. Rev. E6287468752CrossRefGoogle Scholar
  11. Chin, S.A., ChenC.R.,  2001‘Fourth order gradient symplectic integrator methods for solving the time-dependent Schrodinger equation’J. Chem. Phys.11473387341CrossRefGoogle Scholar
  12. Chin, S.A., Chen, C.R. 2002‘Gradient symplectic algorithms for solving the Schrodinger equation with time-dependent potentials’J. Chem. Phys.11714091415CrossRefGoogle Scholar
  13. Chin, S.A. 2004‘Quantum statistical calculations and symplectic corrector algorithms’Phys. Rev. E69046118CrossRefGoogle Scholar
  14. Ciftja, O., Chin, S.A. 2003‘Short-time-evolved wave functions for solving quantum many-body problems’Phys. Rev. B68134510CrossRefGoogle Scholar
  15. Creutz, M., Gocksch, A. 1989‘Higher-order hydrid Monte-Carlo algorithms’Phys. Rev. Lett.63912CrossRefPubMedGoogle Scholar
  16. Forbert, H.A., Chin, S.A. 2001‘Fourth-order algorithms for solving the multivariable Langevin equation and the Kramers equation’Phys. Rev. E63016703CrossRefGoogle Scholar
  17. Forbert, H.A., Chin, S.A. 2001‘Fourth-order diffusion Monte Carlo algorithms for solving quantum many-body problems’Phys. Rev. B63144518CrossRefGoogle Scholar
  18. Forest, E., Ruth, R.D. 1990‘4th-order symplectic integration’Physica D43105117CrossRefGoogle Scholar
  19. Gladman, B., Duncan, M., Candy, J. 1991‘Symplectic integrators for long-term integrations in celestial mechanics’Celest. Mech. Dyn. Astron.52221240CrossRefGoogle Scholar
  20. Goldman, D., Kaper, T.J. 1996‘Nth-order operator splitting schemes and nonreversible systems’SIAM J. Numer. Anal.33349367CrossRefGoogle Scholar
  21. Jang, S., Jang, S., Voth, G.A. 2001‘Applications of higher order composite factorization schemes in imaginary time path integral simulations’J. Chem. Phys.11578327842CrossRefGoogle Scholar
  22. Kinoshita, H., Yoshida, H., Nakai, H. 1991‘Symplectic integrators and their application to dynamical astronomy’Celest. Mech. Dyn. Astron.505971CrossRefGoogle Scholar
  23. Koseleff P.V. (1993). in Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993). Lecture Notes in Comput. Sci., Vol. 673, Springer, Berlin, p. 213Google Scholar
  24. Koseleff P.V. (1996). In Integration algorithms and classical mechanics, Fields Inst. Commun., 10, Amer. Math. Soc., Providence, RI, p. 103Google Scholar
  25. Laskar, J., Robutel, P. 2001‘Higher order symplectic integrators for perturbed Hamiltonian systems’Celest. Mech. Dyn. Astr.803962CrossRefGoogle Scholar
  26. Lopez-Marcos, M.A., Sanz-Serna, J.M., Skeel, R.D. 1996In: DF., GriffithsG. A., Watson eds. Numerical Analysis 1995.LongmanHarlow, UK107122Google Scholar
  27. Lopez-Marcos, M.A., Sanz-Serna, J.M., Skeel, R.D. 1997‘Explicit symplectic integrators using Hessian-vector products’SIAM J. Sci. Comput.18223238CrossRefGoogle Scholar
  28. McLachlan, R.I., Atela, P. 1991‘The accuracy of symplectic integrators’ Nonlinearity.5541562Google Scholar
  29. McLachlan, R.I. 1995a‘On the numerical integration of ordinary differential equations by symmetric composition methods’SIAM J. Sci. Comput.16151168CrossRefGoogle Scholar
  30. McLachlan, R.I. 1995b‘Composition methods in the presence of a small parameters’BIT35258268CrossRefGoogle Scholar
  31. McLachlan, R.I. 1996‘More on symplectic correctors’Marsden, J.E.Patrick, G.W.Shadwick, W.F. eds. Integration Algorithms and Classical Mechanics.American Mathematical SocietyProvidence, RI141149Google Scholar
  32. McLachlan R.I., Scovel C. (1996). ‘Open problems in symplectic integration’. ibid, pp. 151–180Google Scholar
  33. McLachlan, R.I., Reinout, G., Quispel, W. 2002‘Splitting methods’Acta Numerica.11241434CrossRefGoogle Scholar
  34. Omelyan, I.P., Mryglod, I.M., Folk, R. 2002‘Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems’Phys. Rev. E66026701CrossRefGoogle Scholar
  35. Omelyan, I.P., Mryglod, I.M., Folk, R. 2003‘Symplectic analytically integrable decomposition algorithms: Classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations’Comput. Phys. Commun.151271314Google Scholar
  36. Rowlands, G. 1991‘A numerical algorithm for Hamiltonian-systems’J. Comput. Phys.97235239CrossRefGoogle Scholar
  37. Ruth, R. 1983‘A canonical integration technique’IEEE Trans. Nucl. Sci.3026692671Google Scholar
  38. Scuro S., Chin S.A. (2004). Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motion, arXiv, math-ph/0411086Google Scholar
  39. Sheng, Q. 1989‘Solving linear partial differential equations by exponential splitting’IMA J. Numer. Anal.9199212Google Scholar
  40. Suzuki, M. 1990‘Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations’Phys. Lett. A146319323CrossRefGoogle Scholar
  41. Suzuki, M. 1991‘General theory of fractal path-integrals with applications to many-body theories and statistical physics’J. Math. Phys.32400407CrossRefGoogle Scholar
  42. Suzuki, M. 1993‘General decomposition-theory of ordered exponentials’Proc. Jpn. Acad. Ser. B69161166Google Scholar
  43. Suzuki, M. 1996‘New scheme of hybrid Exponential product formulas with applications to quantum Monte Carlo simulations’Landau, D.Mon, K.Shuttler, H. eds. Computer Simulation Studies in Condensed Matter Physics VIII.SpringerBerlin16Google Scholar
  44. Wisdom, J., Holman, M. 1991‘Symplectic maps for the N-body problem’Astron. J.10215281538CrossRefGoogle Scholar
  45. Wisdom, J., Holman, M., Touma, J. 1996‘Symplectic correctors’Marsden, J.E.Patrick, G.W.Shadwick, W.F. eds. Integration Algorithms and Classical Mechanics.American Mathematical SocietyProvidence, RI217244Google Scholar
  46. Yoshida, H. 1990‘Construction of higher order symplectic integrators’Phys. Lett. A150262268CrossRefGoogle Scholar
  47. Yoshida, H. 1993‘Recent progress in the theory and application of symplectic integrators’Celest. Mech. Dyn. Astron.562743CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Center for Theoretical Physics, Department of PhysicsTexas A & M UniversityTX

Personalised recommendations