Skip to main content
SpringerLink
Log in

General techniques for constructing variational integrators

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton-Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benettin G, Giorgilli A. On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J Stat Phys, 1994, 74: 1117–1143

    Article  MathSciNet  MATH  Google Scholar 

  2. Bou-Rabee N, Owhadi H. Stochastic variational integrators. IMA J Numer Anal, 2009, 29(2): 421–443

    Article  MathSciNet  MATH  Google Scholar 

  3. Cortés J, Martínez S. Non-holonomic integrators. Nonlinearity, 2001, 14(5): 1365–1392

    Article  MathSciNet  MATH  Google Scholar 

  4. Cuell C, Patrick G. Geometric discrete analogues of tangent bundles and constrained Lagrangian systems. J Geom Phys, 2009, 59(7): 976–997

    Article  MathSciNet  MATH  Google Scholar 

  5. Fetecau R, Marsden J, Ortiz M, West M. Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM Journal on Applied Dynamical Systems, 2003, 2(3): 381–416

    Article  MathSciNet  MATH  Google Scholar 

  6. Hairer E. Backward analysis of numerical integrators and symplectic methods. Scientific Computation and Differential Equations (Auckland, 1993). Ann Numer Math, 1994, 1(1–4): 107–132

    MathSciNet  MATH  Google Scholar 

  7. Hairer E, Lubich C. The life-span of backward error analysis for numerical integrators. Numer Math, 1997, 76: 441–462

    Article  MathSciNet  MATH  Google Scholar 

  8. Hairer E, Lubich C, Wanner G. Geometric Numerical Integration. 2nd ed. Springer Series in Computational Mathematics, Vol 31. Berlin: Springer-Verlag, 2006

    MATH  Google Scholar 

  9. Iserles A, Munthe-Kaas H, Nørsett S, Zanna A. Lie-group methods. In: Acta Numerica, Vol 9. Cambridge: Cambridge University Press, 2000, 215–365

    Google Scholar 

  10. Kahan W. Further remarks on reducing truncation errors. Commun ACM, 1965, 8: 40

    Article  Google Scholar 

  11. Keller H B. Numerical methods for two-point boundary value problems. New York: Dover Publications Inc, 1992

    Google Scholar 

  12. Lall S, West M. Discrete variational Hamiltonian mechanics. J Phys A, 2006, 39(19): 5509–5519

    Article  MathSciNet  MATH  Google Scholar 

  13. Lee T, Leok M, McClamroch N. Lie group variational integrators for the full body problem. Comput Methods Appl Mech Engrg, 2007, 196(29–30): 2907–2924

    Article  MathSciNet  MATH  Google Scholar 

  14. Lee T, Leok M, McClamroch N. Lie group variational integrators for the full body problem in orbital mechanics. Celestial Mech Dynam Astronom, 2007, 98(2): 121–144

    Article  MathSciNet  MATH  Google Scholar 

  15. Lee T, Leok M, McClamroch N. Lagrangian mechanics and variational integrators on two-spheres. Int J Numer Methods Eng, 2009, 79(9): 1147–1174

    Article  MathSciNet  MATH  Google Scholar 

  16. Leok M. Generalized Galerkin variational integrators: Lie group, multiscale, and pseudospectral methods. Preprint, 2004, arXiv: math.NA/0508360

  17. Leok M, Shingel T. Prolongation-collocation variational integrators. IMA J Numer Anal (in press), arXiv: 1101.1995 [math.NA]

  18. Leok M, Zhang J. Discrete Hamiltonian variational integrators. IMA J Numer Anal, 2011, 31(4): 1497–1532

    Article  MathSciNet  MATH  Google Scholar 

  19. Lew A, Marsden J E, Ortiz M, West M. Asynchronous variational integrators. Arch Ration Mech Anal, 2003, 167(2): 85–146

    Article  MathSciNet  MATH  Google Scholar 

  20. Leyendecker S, Marsden J, Ortiz M. Variational integrators for constrained mechanical systems. Z Angew Math Mech, 2008, 88: 677–708

    Article  MathSciNet  MATH  Google Scholar 

  21. Marsden J, Pekarsky S, Shkoller S. Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity, 1999, 12(6): 1647–1662

    Article  MathSciNet  MATH  Google Scholar 

  22. Marsden J E, West M. Discrete mechanics and variational integrators. Acta Numer, 2001, 10: 357–514

    Article  MathSciNet  MATH  Google Scholar 

  23. Oliver M, West M, Wulff C. Approximate momentum conservation for spatial semidiscretizations of nonlinear wave equations. Numer Math, 2004, 97: 493–535

    Article  MathSciNet  MATH  Google Scholar 

  24. Patrick G, Spiteri R, Zhang W, Cuell C. On converting any one-step method to a variational integrator of the same order. In: 7th International Conference on Multibody systems, Nonlinear Dynamics, and Control, Vol 4. 2009, 341–349

    Google Scholar 

  25. Reich S. Backward error analysis for numerical integrators. SIAM J Numer Anal, 1999, 36: 1549–1570

    Article  MathSciNet  MATH  Google Scholar 

  26. Stern A, Grinspun E. Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model Simul, 2009, 7(4): 1779–1794

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093, USA

    Melvin Leok & Tatiana Shingel

Authors
  1. Melvin Leok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatiana Shingel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Melvin Leok.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leok, M., Shingel, T. General techniques for constructing variational integrators. Front. Math. China 7, 273–303 (2012). https://doi.org/10.1007/s11464-012-0190-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-012-0190-9

Keywords

MSC

Search

Navigation