On the Γ-convergence of discrete dynamics and variational integrators


For a simple class of Lagrangians and variational integrators, derived by time discretization of the action functional, we establish (i) the Γ-convergence of the discrete action sum to the action functional; (ii) the relation between Γ-convergence and weak* convergence of the discrete trajectories in {itW{su1,℞}}({ofR};{ofr{sun}; and (iii) the relation between Γ-convergence and the convergence of the Fourier transform of the discrete trajectories as measured in the flat norm.

This is a preview of subscription content, access via your institution.


  1. [1]

    Belytschko, T., and R. Mullen [1976], Mesh partitions of explicit-implicit time integrators. In K.-J. Bathe, J. T. Oden, and W. Wunderlich, editors,Formulations and Computational Algorithms in Finite Element Analysis, 673–690. MIT Press, Cambridge, Mass.

    Google Scholar 

  2. [2]

    Belytschko, T. [1981], Partitioned and adaptive algorithms for explicit time integration. In W. Wunderlich, E. Stein, and K.-J. Bathe, editors,Nonlinear Finite Element Analysis in Structural Mechanics, 572–584. Springer-Verlag, New York.

    Google Scholar 

  3. [3]

    DalMaso, G. [1993], Anintroduction to Γ-convergence. Birkhäuser, Boston-Basel-Berlin.

    Google Scholar 

  4. [4]

    DeGiorgi, E., and T. Franzoni [1975], Su un tipo di convergenza variazionale,Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8),58, 841–850.

    Google Scholar 

  5. [5]

    Hairer, E., and C. Lubich [1997], The life-span of backward error analysis for numerical integrators,Numerische Mathematik,76, 441–462.

    Article  MATH  Google Scholar 

  6. [6]

    Hughes, T. J. R. [1987],The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs, N.J.

    MATH  Google Scholar 

  7. [7]

    Kane, C, J. E. Marsden, and M. Ortiz [1999], Symplectic energy-momentum integrators,J. Math.Phys.,40, 3353–3371.

    Article  MATH  Google Scholar 

  8. [8]

    Maggi, F., and M. Morini [2003], A Γ-convergence result for variational integrators of quadratic Lagrangians, Preprint 03-CNA-008, Carnegie Mellon University.

  9. [9]

    Marsden, J. E., G. W. Patrick, and S. Shkoller [1998], Multisymplectic geometry, variational integrators and nonlinear PDEs,Commun. Math. Phys.,199, 351–395.

    Article  MATH  Google Scholar 

  10. [10]

    Marsden, J. E. and M. West [2001], Discrete variational mechanics and variational integrators,Acta Numerica,10, 357–514.

    Article  MATH  Google Scholar 

  11. [11]

    Moser, J., and A. P. Veselov [1991], Discrete versions of some classical integrable systems and factorization of matrix polynomials,Commun. Math. Phys.,139, 217–243.

    Article  MATH  Google Scholar 

  12. [12]

    Reich, S. [1999], Backward error analysis for numerical integrators,SIAM J. Num. Anal.,36, 1549–1570.

    Article  MATH  Google Scholar 

  13. [13]

    Veselov, A. P. [1988], Integrable discrete-time systems and difference operators,Fund. Anal. & Appl.,22, 83–93.

    Article  MATH  Google Scholar 

  14. [14]

    Wendlandt, J. M., and J. E. Marsden [1997], Mechanical integrators derived from a discrete variational principle,Physica D,106, 223–246.

    Article  MATH  Google Scholar 

Download references

Author information



Additional information

Communicated by J. E. Marsden and R. Kohn

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Müller, S., Ortiz, M. On the Γ-convergence of discrete dynamics and variational integrators. J Nonlinear Sci 14, 279–296 (2004). https://doi.org/10.1007/BF02666023

Download citation

Key words

  • discrete dynamics
  • variational integrators
  • Γ-convergence
  • spectral convergence
  • flat norm