# American Institute of Mathematical Sciences

March  2019, 11(1): 1-22. doi: 10.3934/jgm.2019001

## Modified equations for variational integrators applied to Lagrangians linear in velocities

 Technische Universität Berlin, Institut für Mathematik, MA 7-1, Str. des 17. Juni 136, 10623 Berlin, Germany

Received  November 2017 Revised  December 2018 Published  January 2019

Fund Project: This research was supported by the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics".

Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.

Citation: Mats Vermeeren. Modified equations for variational integrators applied to Lagrangians linear in velocities. Journal of Geometric Mechanics, 2019, 11 (1) : 1-22. doi: 10.3934/jgm.2019001
##### References:
 [1] P. Chartier, E. Hairer and G. Vilmart, Numerical integrators based on modified differential equations, Mathematics of computation, 76 (2007), 1941-1953.  doi: 10.1090/S0025-5718-07-01967-9. [2] G. De La Torre and T. D. Murphey, On the benefits of surrogate lagrangians in optimal control and planning algorithms, in Decision and Control, 55th Conference on, IEEE, 2016, 7384-7391. [3] G. De La Torre and T. D. Murphey, Surrogate lagrangians for variational integrators: High order convergence with low order schemes, preprint, arXiv: 1709.03883. [4] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963. [5] H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley Pub. Co., Reading MA, etc., 1980. [6] E. Hairer, Backward error analysis for multistep methods, Numerische Mathematik, 84 (1999), 199-232.  doi: 10.1007/s002110050469. [7] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, etc., 2006. [8] A. Ilchmann, D. H. Owens and D. Prätzel-Wolters, Sufficient conditions for stability of linear time-varying systems, Control Letters, 9 (1987), 157-163.  doi: 10.1016/0167-6911(87)90022-3. [9] M. Leok and J. Zhang, Discrete Hamiltonian variational integrators, IMA Journal of Numerical Analysis, 31 (2011), 1497-1532.  doi: 10.1093/imanum/drq027. [10] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica 2001, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [11] P. K. Newton, The N-Vortex Problem: Analytical Techniques, vol. 145, Springer, New York, etc., 2001. doi: 10.1007/978-1-4684-9290-3. [12] H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste, vol. Ⅲ, Gauthier-Villars, Paris, 1987. [13] C. W. Rowley and J. E. Marsden, Variational integrators for degenerate Lagrangians, with application to point vortices, in Decision and Control, 41st Conference on, IEEE, 2002, 1521-1527. [14] R. Skoog and C. Lau, Instability of slowly varying systems, IEEE Transactions on Automatic Control, 17 (1972), 86-92.  doi: 10.1109/tac.1972.1099866. [15] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Springer, New York, etc., 1999. [16] T. M. Tyranowski and M. Desbrun, Variational partitioned Runge-Kutta methods for Lagrangians linear in velocities, preprint, arXiv: 1401.7904. [17] M. Vermeeren, Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.  doi: 10.1007/s00211-017-0896-4.

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##### References:
 [1] P. Chartier, E. Hairer and G. Vilmart, Numerical integrators based on modified differential equations, Mathematics of computation, 76 (2007), 1941-1953.  doi: 10.1090/S0025-5718-07-01967-9. [2] G. De La Torre and T. D. Murphey, On the benefits of surrogate lagrangians in optimal control and planning algorithms, in Decision and Control, 55th Conference on, IEEE, 2016, 7384-7391. [3] G. De La Torre and T. D. Murphey, Surrogate lagrangians for variational integrators: High order convergence with low order schemes, preprint, arXiv: 1709.03883. [4] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, 1963. [5] H. Goldstein, Classical Mechanics, 2nd edition, Addison-Wesley Pub. Co., Reading MA, etc., 1980. [6] E. Hairer, Backward error analysis for multistep methods, Numerische Mathematik, 84 (1999), 199-232.  doi: 10.1007/s002110050469. [7] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, etc., 2006. [8] A. Ilchmann, D. H. Owens and D. Prätzel-Wolters, Sufficient conditions for stability of linear time-varying systems, Control Letters, 9 (1987), 157-163.  doi: 10.1016/0167-6911(87)90022-3. [9] M. Leok and J. Zhang, Discrete Hamiltonian variational integrators, IMA Journal of Numerical Analysis, 31 (2011), 1497-1532.  doi: 10.1093/imanum/drq027. [10] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica 2001, 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [11] P. K. Newton, The N-Vortex Problem: Analytical Techniques, vol. 145, Springer, New York, etc., 2001. doi: 10.1007/978-1-4684-9290-3. [12] H. Poincaré, Les Méthodes Nouvelles de la Méchanique Céleste, vol. Ⅲ, Gauthier-Villars, Paris, 1987. [13] C. W. Rowley and J. E. Marsden, Variational integrators for degenerate Lagrangians, with application to point vortices, in Decision and Control, 41st Conference on, IEEE, 2002, 1521-1527. [14] R. Skoog and C. Lau, Instability of slowly varying systems, IEEE Transactions on Automatic Control, 17 (1972), 86-92.  doi: 10.1109/tac.1972.1099866. [15] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Springer, New York, etc., 1999. [16] T. M. Tyranowski and M. Desbrun, Variational partitioned Runge-Kutta methods for Lagrangians linear in velocities, preprint, arXiv: 1401.7904. [17] M. Vermeeren, Modified equations for variational integrators, Numerische Mathematik, 137 (2017), 1001-1037.  doi: 10.1007/s00211-017-0896-4.
Pendulum with midpoint rule (left) and trapezoidal rule (right), both with step size $h = 0.35$ and initial point $(3,0)$ (top) and $(1.5,0)$ (bottom).
Dashed curve: exact solution.
Bullets: discrete solution.
Solid curve: solution of the principal modified equation, truncated after second order.
Line segments: visualization of parasitic oscillations
Leapfrogging vortex pairs with the midpoint rule. No parasitic behavior is visible
Leapfrogging vortex pairs with the trapezoidal rule. One observes parasitic oscillations
Enlarged versions of the right hand sections of Figures 2-3: midpoint rule (left) and trapezoidal rule (right)
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