Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Symplectic integration of constrained Hamiltonian systems

Authors: B. Leimkuhler and S. Reich
Journal: Math. Comp. 63 (1994), 589-605
MSC: Primary 65L05; Secondary 34A50, 58F05, 70-08, 70H05
MathSciNet review: 1250772
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A Hamiltonian system in potential form $ (H(q,p) = {p^t}{M^{ - 1}}p/2 + F(q))$ subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in $ {{\mathbf{R}}^n}$. In this paper, methods which reduce "Hamiltonian differential-algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parametrizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint-invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.

References [Enhancements On Off] (What's this?)

  • [1] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
  • [2] V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
  • [3] Uri M. Ascher and Linda R. Petzold, Stability of computational methods for constrained dynamics systems, SIAM J. Sci. Comput. 14 (1993), no. 1, 95–120. MR 1201313,
  • [4] J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Comput. Methods Appl. Mech. Engrg. 1 (1972), 1–16. MR 0391628,
  • [5] K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical solution of initial value problems in differential-algebraic equations, North-Holland Publishing Co., New York, 1989. MR 1101809
  • [6] G. J. Cooper, Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal. 7 (1987), no. 1, 1–13. MR 967831,
  • [7] R. Courant and D. Hilbert, Methods of mathematical physics, Wiley, New York, 1953.
  • [8] P. A. M. Dirac, Lectures on quantum mechanics, Belfer Graduate School Monographs, no. 3, Yeshiva University, 1964.
  • [9] E. Eich, C. Führer, B. Leimkuhler, and S. Reich, Stabilization and projection methods for multibody dynamics, Report A281, Helsinki Univ. of Technology, Helsinki, 1990.
  • [10] L. Jay, Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, Technical Report, Université de Genève, 1993.
  • [11] B. Leimkuhler and S. Reich, Numerical methods for constrained Hamiltonian systems, Technical Report, Konrad Zuse Center, Berlin, 1992.
  • [12] B. Leimkuhler and R. D. Skeel, Symplectic numerical integrators for constrained molecular dynamics, Technical Report, Dept. of Math., University of Kansas, Lawrence KS, 1992.
  • [13] N. Harris McClamroch and Anthony M. Bloch, Control of constrained Hamiltonian systems and applications to control of constrained robots, Dynamical systems approaches to nonlinear problems in systems and circuits (Henniker, NH, 1986) SIAM, Philadelphia, PA, 1988, pp. 394–403. MR 970064
  • [14] T. Mrziglod, Zur Theorie und numerischen Realisierung von Lösungsmethoden bei Differentialgleichungen mit angekoppelten algebraischen Gleichungen, Diplomarbeit, Math. Inst., Univ. zu Köln, 1987.
  • [15] J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta numerica, 1992, Acta Numer., Cambridge Univ. Press, Cambridge, 1992, pp. 243–286. MR 1165727
  • [16] Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734
  • [17] Florian A. Potra and Werner C. Rheinboldt, On the numerical solution of Euler-Lagrange equations, Mech. Structures Mach. 19 (1991), no. 1, 1–18. MR 1142054,
  • [18] S. Reich, Symplectic integration of constrained Hamiltonian systems by Runge-Kutta methods, Technical Report 93-13, Dept. of Comput. Sci., University of British Columbia, 1993.
  • [19] Werner C. Rheinboldt, Numerical analysis of parametrized nonlinear equations, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 7, John Wiley & Sons, Inc., New York, 1986. A Wiley-Interscience Publication. MR 815107
  • [20] J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT 28 (1988), no. 4, 877–883. MR 972812,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L05, 34A50, 58F05, 70-08, 70H05

Retrieve articles in all journals with MSC: 65L05, 34A50, 58F05, 70-08, 70H05

Additional Information

Keywords: Differential-algebraic equations, constrained Hamiltonian systems, canonical discretization schemes, symplectic methods
Article copyright: © Copyright 1994 American Mathematical Society