Journal of Statistical Physics

, Volume 61, Issue 5–6, pp 1203–1222 | Cite as

Hamilton's equations for constrained dynamical systems

  • Simon W. de Leeuw
  • John W. Perram
  • Henrik G. Petersen
Articles

Abstract

We derive expressions for the conjugate momenta and the Hamiltonian for classical dynamical systems subject to holonomic constraints. We give an algorithm for correcting deviations of the constraints arising in numerical solution of the equations of motion. We obtain an explicit expression for the momentum integral for constrained systems.

Key words

Constraint dynamics Hamiltonian metric tensor 

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References

  1. 1.
    J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics ofn-alkanes,J. Comp. Phys. 23:327 (1977).Google Scholar
  2. 2.
    J. Orban and J. P. Ryckaert, unpublished (1974) [cited in ref. 1].Google Scholar
  3. 3.
    G. Ciccotti and J. P. Ryckaert, Molecular dynamics simulation of rigid molecules,Comp. Phys. Rep. 4:345 (1986).Google Scholar
  4. 4.
    R. Edberg, D. J. Evans, and G. P. Morriss, Constrained molecular dynamics: Simulation of liquid alkanes with a new algorithm,J. Chem. Phys. 84:6933 (1986).Google Scholar
  5. 5.
    M. P. Allen and D. J. Tildesley,Computer Simulation of Liquids (Oxford University Press, 1987).Google Scholar
  6. 6.
    J. W. Perram and H. G. Petersen, New rigid body equations of motion for molecular dynamics,Mol. Simul. 1:239 (1988).Google Scholar
  7. 7.
    J. W. Perram and H. G. Petersen, Algorithms for computing the dynamical trajectories of flexible bodies,Mol. Phys. 65:861 (1988).Google Scholar
  8. 8.
    C. Lanczos,The Variational Principles of Mechanics (University of Toronto Press, 1949).Google Scholar
  9. 9.
    R. Weinstock,Calculus of Variations (McGraw-Hill, 1952).Google Scholar
  10. 10.
    G. Gallivotti,The Elements of Mechanics (Springer-Verlag, 1983).Google Scholar
  11. 11.
    M. Fixman, Classical statistical mechanics of constraints: A theorem and application to polymers,Proc. Natl. Acad. Sci. USA 71:3050 (1974).Google Scholar
  12. 12.
    C. J. Thompson,Mathematical Statistical Mechanics (Macmillan, 1972).Google Scholar
  13. 13.
    E. T. Whittaker,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, 1917).Google Scholar
  14. 14.
    H. Goldstein,Classical Mechanics (Addison-Wesley, 1950).Google Scholar
  15. 15.
    E. C. D. Sudarshan and N. Mukunda,Classical Dynamics: A Modern Perspective (Wiley, 1974).Google Scholar
  16. 16.
    P. A. M. Dirac, Generalized Hamiltonian dynamics,Can. J. Math. 2:129 (1950).Google Scholar
  17. 17.
    P. A. M. Dirac, Generalized Hamiltonian dynamics,Proc. R. Soc. A 246:326 (1958).Google Scholar
  18. 18.
    J. L. Anderson and P. G. Bergmann, Constraints in covariant field theories,Phys. Rev. 83:1018 (1951).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • Simon W. de Leeuw
    • 1
  • John W. Perram
    • 2
  • Henrik G. Petersen
    • 2
  1. 1.Laboratory for Physical ChemistryUniversity of AmsterdamWS AmsterdamThe Netherlands
  2. 2.Department of Mathematics and Computer ScienceOdense UniversityOdense MDenmark

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