Frontiers of Mathematics in China

, Volume 7, Issue 2, pp 273–303

General techniques for constructing variational integrators


    • Department of MathematicsUniversity of California
  • Tatiana Shingel
    • Department of MathematicsUniversity of California
Research Article

DOI: 10.1007/s11464-012-0190-9

Cite this article as:
Leok, M. & Shingel, T. Front. Math. China (2012) 7: 273. doi:10.1007/s11464-012-0190-9


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.


Geometric numerical integrationgeometric mechanicssymplectic integratorvariational integratorLagrangian mechanics



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© Higher Education Press and Springer-Verlag Berlin Heidelberg 2012