Foundations of Computational Mathematics

, Volume 14, Issue 2, pp 339–370 | Cite as

Geometric Generalisations of shake and rattle

  • Robert I McLachlan
  • Klas Modin
  • Olivier Verdier
  • Matt Wilkins
Article

Abstract

A geometric analysis of the shake and rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises shake and rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting.

In order for shake and rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.

Keywords

Symplectic integrators Constrained Hamiltonian systems Coisotropic submanifolds Differential algebraic equations 

Mathematics Subject Classification (2010)

37M15 65P10 70H45 65L80 

References

  1. 1.
    H.C. Andersen, Rattle: a “velocity” version of the shake algorithm for molecular dynamics calculations, J. Comput. Phys. 52, 24–34 (1983). CrossRefMATHGoogle Scholar
  2. 2.
    P. Dirac, Lectures on Quantum Mechanics (Dover Publications, New York, 2001). Google Scholar
  3. 3.
    Z. Ge, J.E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A 133, 134–139 (1988). CrossRefMathSciNetGoogle Scholar
  4. 4.
    M.J. Gotay, On coisotropic imbeddings of presymplectic manifolds, Proc. Am. Math. Soc. 84, 111–114 (1982). CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    M. Gotay, J. Nester, G. Hinds, Presymplectic manifolds and the Dirac–Bergmann theory of constraints, J. Math. Phys. 19, 2388 (1978). CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge University Press, Cambridge, 1990). MATHGoogle Scholar
  7. 7.
    E. Hairer, S. Nørsett, G. Wanner, Solving Ordinary Differential Equations: Nonstiff Problems (Springer, Berlin, 1993). MATHGoogle Scholar
  8. 8.
    E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer, Berlin, 2006). Google Scholar
  9. 9.
    L. Jay, Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, SIAM J. Numer. Anal. 33, 368–387 (1996). CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics (Cambridge University Press, Cambridge, 2004). MATHGoogle Scholar
  11. 11.
    B. Leimkuhler, R. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems, J. Comput. Phys. 112, 117–125 (1994). CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    P. Libermann, C. Marle, Symplectic Geometry and Analytical Mechanics (D. Reidel Publishing Co., Dordrecht, 1987). CrossRefMATHGoogle Scholar
  13. 13.
    J. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (Springer, New York, 1999). CrossRefMATHGoogle Scholar
  14. 14.
    R. McLachlan, K. Modin, O. Verdier, M. Wilkins, Symplectic integrators for index one constraints (2012). arXiv:1207.4250.
  15. 15.
    S. Reich, Symplectic integration of constrained Hamiltonian systems by composition methods, SIAM J. Numer. Anal. 33, 475–491 (1996). CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    J.-P. Ryckaert, G. Ciccotti, H.J. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes, J. Comput. Phys. 23, 327–341 (1977). CrossRefGoogle Scholar

Copyright information

© SFoCM 2013

Authors and Affiliations

  • Robert I McLachlan
    • 1
  • Klas Modin
    • 2
  • Olivier Verdier
    • 3
  • Matt Wilkins
    • 1
  1. 1.Institute of Fundamental SciencesMassey UniversityPalmersston NorthNew Zealand
  2. 2.Department of Mathematical SciencesChalmers University of TechnologyGothenburgSweden
  3. 3.Department of MathematicsNTNUTrondheimNorway

Personalised recommendations