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.
Similar content being viewed by others
References
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
Bou-Rabee N, Owhadi H. Stochastic variational integrators. IMA J Numer Anal, 2009, 29(2): 421–443
Cortés J, Martínez S. Non-holonomic integrators. Nonlinearity, 2001, 14(5): 1365–1392
Cuell C, Patrick G. Geometric discrete analogues of tangent bundles and constrained Lagrangian systems. J Geom Phys, 2009, 59(7): 976–997
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
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
Hairer E, Lubich C. The life-span of backward error analysis for numerical integrators. Numer Math, 1997, 76: 441–462
Hairer E, Lubich C, Wanner G. Geometric Numerical Integration. 2nd ed. Springer Series in Computational Mathematics, Vol 31. Berlin: Springer-Verlag, 2006
Iserles A, Munthe-Kaas H, Nørsett S, Zanna A. Lie-group methods. In: Acta Numerica, Vol 9. Cambridge: Cambridge University Press, 2000, 215–365
Kahan W. Further remarks on reducing truncation errors. Commun ACM, 1965, 8: 40
Keller H B. Numerical methods for two-point boundary value problems. New York: Dover Publications Inc, 1992
Lall S, West M. Discrete variational Hamiltonian mechanics. J Phys A, 2006, 39(19): 5509–5519
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
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
Lee T, Leok M, McClamroch N. Lagrangian mechanics and variational integrators on two-spheres. Int J Numer Methods Eng, 2009, 79(9): 1147–1174
Leok M. Generalized Galerkin variational integrators: Lie group, multiscale, and pseudospectral methods. Preprint, 2004, arXiv: math.NA/0508360
Leok M, Shingel T. Prolongation-collocation variational integrators. IMA J Numer Anal (in press), arXiv: 1101.1995 [math.NA]
Leok M, Zhang J. Discrete Hamiltonian variational integrators. IMA J Numer Anal, 2011, 31(4): 1497–1532
Lew A, Marsden J E, Ortiz M, West M. Asynchronous variational integrators. Arch Ration Mech Anal, 2003, 167(2): 85–146
Leyendecker S, Marsden J, Ortiz M. Variational integrators for constrained mechanical systems. Z Angew Math Mech, 2008, 88: 677–708
Marsden J, Pekarsky S, Shkoller S. Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity, 1999, 12(6): 1647–1662
Marsden J E, West M. Discrete mechanics and variational integrators. Acta Numer, 2001, 10: 357–514
Oliver M, West M, Wulff C. Approximate momentum conservation for spatial semidiscretizations of nonlinear wave equations. Numer Math, 2004, 97: 493–535
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
Reich S. Backward error analysis for numerical integrators. SIAM J Numer Anal, 1999, 36: 1549–1570
Stern A, Grinspun E. Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model Simul, 2009, 7(4): 1779–1794
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-012-0190-9
Keywords
- Geometric numerical integration
- geometric mechanics
- symplectic integrator
- variational integrator
- Lagrangian mechanics