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 integration geometric mechanics symplectic integrator variational integrator Lagrangian mechanics


37J10 65P10 70H25

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© Higher Education Press and Springer-Verlag Berlin Heidelberg 2012