Foundations of Computational Mathematics

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

Geometric Generalisations of shake and rattle

  • Robert I McLachlanEmail author
  • Klas Modin
  • Olivier Verdier
  • Matt Wilkins


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.


Symplectic integrators Constrained Hamiltonian systems Coisotropic submanifolds Differential algebraic equations 

Mathematics Subject Classification (2010)

37M15 65P10 70H45 65L80 



O. Verdier would like to acknowledge the support of the GeNuIn Project, funded by the Research Council of Norway, the Marie Curie International Research Staff Exchange Scheme Fellowship within the European Commission’s Seventh Framework Programme as well as the hospitality of the Institute for Fundamental Sciences of Massey University, New Zealand, where some of this research was conducted. K. Modin would like to thank the Marsden Fund in New Zealand, the Department of Mathematics at NTNU in Trondheim, the Royal Swedish Academy of Science and the Swedish Research Council, contract VR-2012-335, for support. We would like to thank the reviewers for helpful suggestions.


  1. 1.
    H.C. Andersen, Rattle: a “velocity” version of the shake algorithm for molecular dynamics calculations, J. Comput. Phys. 52, 24–34 (1983). CrossRefzbMATHGoogle 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). CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    M. Gotay, J. Nester, G. Hinds, Presymplectic manifolds and the Dirac–Bergmann theory of constraints, J. Math. Phys. 19, 2388 (1978). CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge University Press, Cambridge, 1990). zbMATHGoogle Scholar
  7. 7.
    E. Hairer, S. Nørsett, G. Wanner, Solving Ordinary Differential Equations: Nonstiff Problems (Springer, Berlin, 1993). zbMATHGoogle 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). CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics (Cambridge University Press, Cambridge, 2004). zbMATHGoogle Scholar
  11. 11.
    B. Leimkuhler, R. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems, J. Comput. Phys. 112, 117–125 (1994). CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    P. Libermann, C. Marle, Symplectic Geometry and Analytical Mechanics (D. Reidel Publishing Co., Dordrecht, 1987). CrossRefzbMATHGoogle Scholar
  13. 13.
    J. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems (Springer, New York, 1999). CrossRefzbMATHGoogle 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). CrossRefzbMATHMathSciNetGoogle 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
    Email author
  • 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