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Numerical integration
of constrained Hamiltonian systems
using Dirac brackets


Author: Werner M. Seiler
Journal: Math. Comp. 68 (1999), 661-681
MSC (1991): Primary 65L05, 70H05; Secondary 70--08
DOI: https://doi.org/10.1090/S0025-5718-99-01010-8
MathSciNet review: 1604375
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Abstract: We study the numerical properties of the equations of motion of constrained systems derived with Dirac brackets. This formulation is compared with one based on the extended Hamiltonian. As concrete examples, a pendulum in Cartesian coordinates and a chain molecule are treated.


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  • 1. H.C. Andersen, Rattle - a velocity version of the Shake algorithm for molecular dynamics calculations, J. Comp. Phys. 52 (1983), 24-34.
  • 2. V.I. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics 60, Springer-Verlag, New York, 1978. MR 57:14033b
  • 3. U.M. Ascher, H. Chin, and S. Reich, Stabilization of DAEs and invariant manifolds, Num. Math. 67 (1994), 131-149. MR 94m:65121
  • 4. J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comp. Meth. Appl. Mech. Eng. 1 (1972), 1-16. MR 52:12448
  • 5. K.E. Brenan, S.L. Campbell, and L.R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations, Classics in Applied Mathematics 14, SIAM, Philadelphia, 1996. MR 96h:65083
  • 6. G.J. Cooper, Stability of Runge-Kutta methods for trajectory problems, IMA J. Num. Anal. 7 (1987), 1-13. MR 90d:65133
  • 7. D.J. Dichmann, J.H. Maddocks, and R.L. Pego, Hamiltonian dynamics of an elastica and the stability of solitary waves, Arch. Rat. Mech. Anal. 135 (1996), 357-396. MR 97m:58072
  • 8. P.A.M. Dirac, Generalized Hamiltonian dynamics, Can. J. Math. 2 (1950), 129-148. MR 13:306b
  • 9. -, Generalized Hamiltonian dynamics, Proc. Roy. Soc. A 246 (1958), 326-332. MR 20:724
  • 10. G.V. Dunne, R. Jackiw, and C.A. Trugenberger, Topological (Chern-Simons) quantum mechanics, Phys. Rev. D 41 (1990), 661-666. MR 90m:81031
  • 11. H. Goldstein, Classical mechanics, Addison-Wesley, New York, 1980. MR 81j:70001
  • 12. W. Hahn, Stability of motion, Grundlehren der mathematischen Wissenschaften 138, Springer-Verlag, Berlin, 1967. MR 36:6716
  • 13. M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press, 1992. MR 94h:81003
  • 14. L.O. Jay, Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, SIAM J. Numer. Anal. 33 (1996), 368-387. MR 97e:65065
  • 15. B.J. Leimkuhler and S. Reich, Symplectic integration of constrained Hamiltonian systems, Math. Comp. 63 (1994), 589-605. MR 95b:65082
  • 16. B.J. Leimkuhler and R.D. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems, J. Comp. Phys. 112 (1994), 117-125. MR 95h:58053
  • 17. R.S. MacKay, Stability of equilibria of Hamiltonian systems, Nonlinear Phenomena and Chaos (S. Sarkar, ed.), Hilger, Bristol, 1986, pp. 254-270. MR 87j:58063
  • 18. J.H. Maddocks and R.L. Pego, An unconstrained Hamiltonian formulation for incompressible fluid flow, Comm. Math. Phys. 170 (1995), 207-217. MR 96a:76085
  • 19. P.C. Moan, The numerical solution of ordinary differential equations with conservation laws, Master's thesis, Norwegian Institute of Technology, Trondheim, 1996.
  • 20. G.R.W. Quispel and H.W. Capel, Solving ODEs numerically while preserving all first integrals, Preprint, La Trobe University, Melbourne, 1997.
  • 21. G.R.W. Quispel and G.S. Turner, Discrete gradient methods for solving ODEs numerically while preserving a first integral, J. Phys. A 29 (1996), L341-L349. MR 97e:65028
  • 22. S. Reich, Symplectic integration of constrained Hamiltonian systems by composition methods, SIAM J. Num. Anal. 33 (1996), 475-491. MR 97h:65103
  • 23. R. Salmon, Semigeostrophic theory as a Dirac-bracket projection, J. Fluid Mech. 196 (1988), 345-358. MR 90j:76097
  • 24. J.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian problems, Applied Mathematics and Mathematical Computation 7, Chapman & Hall, London, 1994. MR 95f:65006
  • 25. W.M. Seiler, Involution and constrained dynamics II: The Faddeev-Jackiw approach, J. Phys. A 28 (1995), 7315-7331. MR 97h:70018
  • 26. -, Momentum versus position projections for constrained Hamiltonian systems, Num. Algo., submitted.
  • 27. -, Numerical analysis of constrained Hamiltonian systems and the formal theory of differential equations, Math. Comp. Simul. 45 (1998), 561-576.
  • 28. W.M. Seiler and R.W. Tucker, Involution and constrained dynamics I: The Dirac approach, J. Phys. A 28 (1995), 4431-4451. MR 97f:70024
  • 29. C.L. Siegel and J.K. Moser, Lectures on celestial mechanics, Grundlehren der mathematischen Wissenschaften 187, Springer-Verlag, Berlin, 1971. MR 58:19464
  • 30. J. \'{S}niatycki, Dirac brackets in geometric dynamics, Ann. Inst. Henri Poincaré A 20 (1974), 365-372. MR 50:11319
  • 31. K. Sundermeyer, Constrained dynamics, Lecture Notes in Physics 169, Springer-Verlag, New York, 1982. MR 84f:58051

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Additional Information

Werner M. Seiler
Affiliation: Lehrstuhl I für Mathematik, Universität Mannheim, D-68131 Mannheim, Germany
Email: wms@ira.uka.de

DOI: https://doi.org/10.1090/S0025-5718-99-01010-8
Keywords: Constrained Hamiltonian system, Dirac bracket, Hamilton-Dirac equations of motion, extended Hamiltonian, numerical integration
Received by editor(s): August 22, 1996
Received by editor(s) in revised form: March 17, 1997, and July 30, 1997
Additional Notes: This work was supported by the Deutsche Forschungsgemeinschaft.
Article copyright: © Copyright 1999 American Mathematical Society

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