Abstract
In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the formH=T(p)+V(q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abia, L. and Sanz-Serna, J.M.: 1990, ‘Partitioned Runge-Kutta methods for separable Hamiltonian problems’, Universidad de Valladolid, Applied Mathematics and Computation Reports1990/8
Auerbach, S.P. and Friedman, A.: 1991, ‘Long-time behaviour of numerically computed orbits: small and intermediate time step analysis of one-dimensional systems’,J. Comp. Phys. 93, 189–223
Calvo, M.P. and Sanz-Serna, J.M: 1991, ‘Variable steps for symplectic integrators’, Universidad de Valladolid, Applied Mathematics and Computation Reports1991/3
Calvo, M.P. and Sanz-Serna, J.M: 1991a, ‘The development of variable-step symplectic integrators, with application to the two-body problem’, Universidad de Valladolid, Applied Mathematics and Computation Reports1991/9
Candy, J. and Rozmus, W.: 1991, ‘A symplectic integration algorithm for separable Hamiltonian functions’,J. Comp. Phys. 92, 230–256
Channell, P.J. and Scovel, J.C.: 1990, ‘Symplectic integration of Hamiltonian systems’,Nonlinearity 3, 231–259
Dekker, K. and Verwer, J.G.: 1984,Stability of Runge-Kutta methods for stiff nonlinear differential equations, North-Holland
Dragt, A.J. and Finn, J.M.: 1976, ‘Lie series and invariant functions for analytic symplectic maps’,J. Math. Phys. 17, 2215–2227
Dragt, A.J., Neri, F., Rangarajan, G., Douglas, D.R., Healy, L.M. and Ryne, R.D.: 1988, ‘Lie algebraic treatment of linear and nonlinear beam dynamics’,Ann. Rev. Nucl. Part. Sci. 38, 455–496
Eirola, T. and Sanz-Serna, J.M.: 1990, ‘Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods’, Universidad de Valladolid, Applied Mathematics and Computation Reports1990/9
Feng, K. and Qin, M.: 1987, ‘The symplectic methods for the computation of Hamiltonian equations’,Lecture Note in Math. 1297, 1–37
Forest, E.: 1987, ‘Canonical integrators as tracking codes’, SSC Central Design Group Technical ReportSSC-138
Forest, E., Brengtsson, J. and Reusch M.F.: 1991, ‘Application of the Yoshida-Ruth techniques to implicit integration and multi-map explicit integration’,Phys. Lett. A 158, 99–101
Forest, E. and Ruth, R.D.: 1990, ‘Fourth-order symplectic integration’,Physica D 43, 105–117
Friedman, A. and Auerbach, S.P.: 1991, ‘Numerically induced stochasticity’,J. Comp. Phys. 93, 171–188
Ge, Z. and Marsden, J.E.: 1988, ‘Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators’,Phys. Lett. A 133, 134–139
Giacaglia, G.E.O.: 1972,Perturbation methods in non-linear systems, Springer
Gladman, B. and Duncan, M.: 1990, ‘On the fates of minor bodies in the outer solar system’,Astron. J. 100, 1680–1693
Gladman, B., Duncan, M. and Candy, J.: 1991, ‘Symplectic integrators for long-term integrations in celestial mechanics’,Celest. Mech. 52, 221–240
Greene, J.M.: 1979, ‘A method for determining a stochastic transition’,J. Math. Phys. 20, 1183–1201
Gustavson, F.: 1966, ‘On constructing formal integrals of a Hamiltonian system near an equilibrium points’,Astron. J. 71, 670–686
Itoh, T. and Abe, K.: 1988, ‘Hamiltonian-conserving discrete canonical equations based on variational difference quotients’,J. Comp. Phys. 77, 85–102
Itoh, T. and Abe, K.: 1989, ‘Discrete Lagrange's equations and canonical equations based on the principle of least action’,Applied Math. Comp. 29, 161–183
Kinoshita, H. and Nakai, H.: 1991, ‘New methods for long-time numerical integration of planetary orbits’, National Astronomical Observatory of Japan, preprint
Kinoshita, H., Yoshida, H. and Nakai, H.: 1991, ‘Symplectic integrators and their application to dynamical astronomy’,Celest. Mech. 50, 59–71
Lasagni, F.: 1988, ‘Canonical Runge-Kutta methods’,ZAMP 39, 952–953
MacKay, R.S.: 1991, ‘Some aspects of the dynamics and numerics of Hamiltonian systems’, University of Warwick, preprint
McLachlan, R. and Atela, P.: 1991, ‘The accuracy of symplectic integrators’,Nonlinearity 5, 541–562
Mersman, W.A.: 1971, ‘Explicit recursive algorithms for the construction of equivalent canonical transformations’,Celest. Mech. 3, 384–389
Neri, F.: 1987, ‘Lie algebras and canonical integration’, Dept. of Physics, University of Maryland, preprint
Okunbor, D. and Skeel, R.D.: 1991, ‘Explicit canonical methods for Hamiltonian systems’, Dept. of Computer Sci., Univ. of Illinois at Urbana-Champaign, preprint
Okunbor, D. and Skeel, R.D.: 1992, ‘An explicit Runge-Kutta-Nystrom method is canonical if and only if its adjoint is explicit’,SIAM J. Numer. Anal. 29,
Okunbor, D. and Skeel, R.D.: 1992a, ‘Canonical Runge-Kutta-Nystrom methods of order 5 and 6’, Dept. of Computer Sci., Univ. of Illinois at Urbana-Champaign, preprint
Pullin, D.I. and Saffman, P.G.: 1991, ‘Long time symplectic integration, the example of four-vortex motion’,Proc. R. Soc. London A 432, 481–494
Quinlan, G.D. and Toomre, A.: 1991, ‘Resonant instabilities in symmetric multistep methods’, University of Toronto, preprint
Quinlan, G.D. and Tremaine, S.: 1990, ‘Symmetric multistep methods for the numerical integration of planetary orbits’,Astron. J. 100, 1694–1700
Ruth, R.D.: 1983, ‘A canonical integration technique’,IEEE Trans. Nucl. Sci. NS-30, 2669–2671
Saito, S., Sugiura, H. and Mitsui, T.: 1992, ‘Family of symplectic implicit Runge-Kutta formulae’, Dept. of information engineering, Nagoya university, preprint
Sanz-Serna, J.M.: 1988, ‘Runge-Kutta schemes for Hamiltonian systems’,BIT 28, 877–883
Sanz-Serna, J.M.: 1991, ‘Symplectic integrators for Hamiltonian problems: an overview’, Universidad de Valladolid, Applied Mathematics and Computation Reports1991/6
Sanz-Serna, J.M. and Abia, L.: 1991, ‘Order conditions for canonical Runge-Kutta schemes’,SIAM J. Numer. Anal. 28, 1081–1096
Scovel, C: 1991, ‘Symplectic numerical integration of Hamiltonian systems’, inThe geometry of Hamiltonian systems, Ratiu, T. ed., Springer, 463–496
Sussman, G.J. and Wisdom, J.: 1988, ‘Numerical evidence that the motion of Pluto is chaotic’,Science 241, 433–437
Suzuki, M.: 1990, ‘Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations’,Phys. Lett. A 146, 319–323
Suzuki, M.: 1991, ‘General theory of fractal path integrals with applications to many-body theories and statistical physics’,J. Math. Phys. 32, 400–407
Suzuki, M.: 1992, ‘General theory of higher-order decomposition of exponential operators and symplectic integrators’, Dept. of Physics, University of Tokyo, preprint
Varadarajan, V.S.: 1974,Lie groups, Lie algebras and their representation, Prentice-Hall
Wisdom, J. and Holman, M.: 1991, ‘Symplectic maps for the N-body problem’,Astron. J. 102, 1528–1538
Yoshida, H.: 1990, ‘Construction of higher order symplectic integrators’,Phys. Lett. A 150, 262–268
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yoshida, H. Recent progress in the theory and application of symplectic integrators. Celestial Mech Dyn Astr 56, 27–43 (1993). https://doi.org/10.1007/BF00699717
Issue Date:
DOI: https://doi.org/10.1007/BF00699717