# American Institute of Mathematical Sciences

December  2021, 13(4): 533-606. doi: 10.3934/jgm.2021017

## Error analysis of forced discrete mechanical systems

 1 Instituto Balseiro, Universidad Nacional de Cuyo – C.N.E.A., Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP, República Argentina 2 Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819- 0395, Japan 3 Instituto Balseiro, Universidad Nacional de Cuyo – C.N.E.A. and CONICET, Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP, República Argentina

Received  August 2019 Revised  March 2021 Published  December 2021 Early access  August 2021

Fund Project: This research was partially supported by grants from the Universidad Nacional de Cuyo (grants 06/C567 and 06/C574) and CONICET

The purpose of this paper is to perform an error analysis of the variational integrators of mechanical systems subject to external forcing. Essentially, we prove that when a discretization of contact order $r$ of the Lagrangian and force are used, the integrator has the same contact order. Our analysis is performed first for discrete forced mechanical systems defined over $TQ$, where we study the existence of flows, the construction and properties of discrete exact systems and the contact order of the flows (variational integrators) in terms of the contact order of the original systems. Then we use those results to derive the corresponding analysis for the analogous forced systems defined over $Q\times Q$.

Citation: Javier Fernández, Sebastián Elías Graiff Zurita, Sergio Grillo. Error analysis of forced discrete mechanical systems. Journal of Geometric Mechanics, 2021, 13 (4) : 533-606. doi: 10.3934/jgm.2021017
##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2$^{nd}$ edition, Pure and Applied Mathematics, vol. 120, Academic Press Inc., Orlando, FL, 1986. [3] A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7. [4] H. Cendra, S. Ferraro and S. Grillo, Lagrangian reduction of generalized nonholonomic systems, J. Geom. Phys., 58 (2008), 1271-1290.  doi: 10.1016/j.geomphys.2008.05.002. [5] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), 108pp. doi: 10.1090/memo/0722. [6] C. Cuell and G. W. Patrick, Skew critical problems, Regul. Chaotic Dyn., 12 (2007), 589-601.  doi: 10.1134/S1560354707060020. [7] ——, Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.  doi: 10.1016/j.geomphys.2009.04.005. [8] J. C. Marrero, D. Martín de Diego and E. Martínez, On the Exact Discrete Lagrangian Function for Variational Integrators: Theory and Applications, arXiv: 1608.01586, 2016. [9] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [10] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [11] D. Martín de Diego and R. Sato Martín de Almagro, Variational order for forced Lagrangian systems, Nonlinearity, 31 (2018), 3814-3846.  doi: 10.1088/1361-6544/aac5a6. [12] G. W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numer. Math., 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3. [13] W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, Acta Phys. Polon. B, The Infeld Centennial Meeting (Warsaw, 1998), 30 (1999), 2909–2978. [14] H. Whitney, Differentiability of the remainder term in Taylor's formula, Duke Math. J., 10 (1943), 153-158.

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##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2$^{nd}$ edition, Pure and Applied Mathematics, vol. 120, Academic Press Inc., Orlando, FL, 1986. [3] A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7. [4] H. Cendra, S. Ferraro and S. Grillo, Lagrangian reduction of generalized nonholonomic systems, J. Geom. Phys., 58 (2008), 1271-1290.  doi: 10.1016/j.geomphys.2008.05.002. [5] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001), 108pp. doi: 10.1090/memo/0722. [6] C. Cuell and G. W. Patrick, Skew critical problems, Regul. Chaotic Dyn., 12 (2007), 589-601.  doi: 10.1134/S1560354707060020. [7] ——, Geometric discrete analogues of tangent bundles and constrained Lagrangian systems, J. Geom. Phys., 59 (2009), 976-997.  doi: 10.1016/j.geomphys.2009.04.005. [8] J. C. Marrero, D. Martín de Diego and E. Martínez, On the Exact Discrete Lagrangian Function for Variational Integrators: Theory and Applications, arXiv: 1608.01586, 2016. [9] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. [10] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X. [11] D. Martín de Diego and R. Sato Martín de Almagro, Variational order for forced Lagrangian systems, Nonlinearity, 31 (2018), 3814-3846.  doi: 10.1088/1361-6544/aac5a6. [12] G. W. Patrick and C. Cuell, Error analysis of variational integrators of unconstrained Lagrangian systems, Numer. Math., 113 (2009), 243-264.  doi: 10.1007/s00211-009-0245-3. [13] W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, Acta Phys. Polon. B, The Infeld Centennial Meeting (Warsaw, 1998), 30 (1999), 2909–2978. [14] H. Whitney, Differentiability of the remainder term in Taylor's formula, Duke Math. J., 10 (1943), 153-158.
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