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August  2017, 37(8): 4249-4275. doi: 10.3934/dcds.2017182

The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators

 1 Departamento de Matemáticas y Mecánica, IIMAS-UNAM, Apdo. Postal: 20-726, Mexico City, 01000, Mexico 2 Department of Applied Mathematics, University of Waterloo, 200 Univ. Avenue West, N2L 3G1, Waterloo, Canada

* Corresponding author: Luis C. García-Naranjo

Received  October 2016 Revised  March 2017 Published  April 2017

Fund Project: LGN was supported by a Newton Advanced Fellowship from the Royal Society, ref: NA140017

Geometric integrators for nonholonomic systems were introduced by Cortés and Martínez in [4] by proposing a discrete Lagrange-D'Alembert principle. Their approach is based on the definition of a discrete Lagrangian $L_d$ and a discrete constraint space $D_d$. There is no recipe to construct these objects and the performance of the integrator is sensitive to their choice.

Cortés and Martínez [4] claim that choosing $L_d$ and $D_d$ in a consistent manner with respect to a finite difference map is necessary to guarantee an approximation of the continuous flow within a desired order of accuracy. Although this statement is given without proof, similar versions of it have appeared recently in the literature.

We evaluate the importance of the consistency condition by comparing the performance of two different geometric integrators for the nonholonomic Suslov problem, only one of which corresponds to a consistent choice of $L_d$ and $D_d$. We prove that both integrators produce approximations of the same order, and, moreover, that the non-consistent discretisation outperforms the other in numerical experiments and in terms of energy preservation. Our results indicate that the consistency of a discretisation might not be the most relevant feature to consider in the construction of nonholonomic geometric integrators.

Citation: Luis C. garcía-Naranjo, Fernando Jiménez. The geometric discretisation of the Suslov problem: A case study of consistency for nonholonomic integrators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4249-4275. doi: 10.3934/dcds.2017182
References:
 [1] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics; Dynamical Systems Ⅲ, 3rd edition, Encyclopaedia of Mathematical Sciences, 3. Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-642-61551-1. Google Scholar [2] A. M. Bloch, Nonholonomic Mechanics and Control, 2nd edition, Springer-Verlag, New York, 2015. doi: 10.1007/b97376. Google Scholar [3] A. I. Bobenko and Y. B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations and semidirect products, Lett. Math. Phys., 49 (1999), 79-93. doi: 10.1023/A:1007654605901. Google Scholar [4] J. Cortés and S. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. doi: 10.1088/0951-7715/14/5/322. Google Scholar [5] Y. N. Fedorov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics, Amer. Math. Soc. Transl., 168 (1995), 141-171. doi: 10.1090/trans2/168/06. Google Scholar [6] Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241. doi: 10.1088/0951-7715/18/5/017. Google Scholar [7] Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA, 3 (2007), Paper 044, 15pp. doi: 10.3842/SIGMA.2007.044. Google Scholar [8] Y. N. Fedorov, A. J. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259. doi: 10.1088/0951-7715/22/9/009. Google Scholar [9] L. C. García-Naranjo, J. C. Marrero, A. J. Maciejewski and M. Przybylska, The inhomogeneous Suslov problem, Phys. Lett. A, 378 (2014), 2389-2394. doi: 10.1016/j.physleta.2014.06.026. Google Scholar [10] D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 351-397. doi: 10.1007/s00332-007-9012-8. Google Scholar [11] A. Iserles, H. Z. Munthe-Kaas, S. P. Norsett and A. Zanna, Lie-group methods, Acta Numerica., 9 (2000), 215-365. doi: 10.1017/S0962492900002154. Google Scholar [12] F. Jiménez and J. Scheurle, On the discretization of nonholonomic mechanics in $\mathbb{R}^N$, J. Geom. Mech., 7 (2015), 43-80. doi: 10.3934/jgm.2015.7.43. Google Scholar [13] F. Jiménez and J. Scheurle, On the discretization of the Euler-Poincaré-Suslov equations in SO(3), arXiv: 1506.01289. To appear in J. Geom. Mech.Google Scholar [14] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17. SpringerVerlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. Google Scholar [15] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. Google Scholar [16] R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1. Google Scholar [17] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. Google Scholar [18] G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moscow, 1946 (in Russian).Google Scholar [19] A. P. Veselov, Integrable discrete-time systems and difference operators, Funct. Anal. Appl., 22 (1988), 1-13. doi: 10.1007/bf01077598. Google Scholar

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References:
 [1] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics; Dynamical Systems Ⅲ, 3rd edition, Encyclopaedia of Mathematical Sciences, 3. Springer-Verlag, Berlin, 2006. doi: 10.1007/978-3-642-61551-1. Google Scholar [2] A. M. Bloch, Nonholonomic Mechanics and Control, 2nd edition, Springer-Verlag, New York, 2015. doi: 10.1007/b97376. Google Scholar [3] A. I. Bobenko and Y. B. Suris, Discrete Lagrangian reduction, discrete Euler-Poincaré equations and semidirect products, Lett. Math. Phys., 49 (1999), 79-93. doi: 10.1023/A:1007654605901. Google Scholar [4] J. Cortés and S. Martínez, Nonholonomic integrators, Nonlinearity, 14 (2001), 1365-1392. doi: 10.1088/0951-7715/14/5/322. Google Scholar [5] Y. N. Fedorov and V. V. Kozlov, Various aspects of n-dimensional rigid body dynamics, Amer. Math. Soc. Transl., 168 (1995), 141-171. doi: 10.1090/trans2/168/06. Google Scholar [6] Y. N. Fedorov and D. V. Zenkov, Discrete nonholonomic LL systems on Lie groups, Nonlinearity, 18 (2005), 2211-2241. doi: 10.1088/0951-7715/18/5/017. Google Scholar [7] Y. N. Fedorov, A discretization of the nonholonomic Chaplygin sphere problem, SIGMA, 3 (2007), Paper 044, 15pp. doi: 10.3842/SIGMA.2007.044. Google Scholar [8] Y. N. Fedorov, A. J. Maciejewski and M. Przybylska, The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions, Nonlinearity, 22 (2009), 2231-2259. doi: 10.1088/0951-7715/22/9/009. Google Scholar [9] L. C. García-Naranjo, J. C. Marrero, A. J. Maciejewski and M. Przybylska, The inhomogeneous Suslov problem, Phys. Lett. A, 378 (2014), 2389-2394. doi: 10.1016/j.physleta.2014.06.026. Google Scholar [10] D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci., 18 (2008), 351-397. doi: 10.1007/s00332-007-9012-8. Google Scholar [11] A. Iserles, H. Z. Munthe-Kaas, S. P. Norsett and A. Zanna, Lie-group methods, Acta Numerica., 9 (2000), 215-365. doi: 10.1017/S0962492900002154. Google Scholar [12] F. Jiménez and J. Scheurle, On the discretization of nonholonomic mechanics in $\mathbb{R}^N$, J. Geom. Mech., 7 (2015), 43-80. doi: 10.3934/jgm.2015.7.43. Google Scholar [13] F. Jiménez and J. Scheurle, On the discretization of the Euler-Poincaré-Suslov equations in SO(3), arXiv: 1506.01289. To appear in J. Geom. Mech.Google Scholar [14] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Texts in Applied Mathematics, 17. SpringerVerlag, New York, 1999. doi: 10.1007/978-0-387-21792-5. Google Scholar [15] J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X. Google Scholar [16] R. McLachlan and M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci., 16 (2006), 283-328. doi: 10.1007/s00332-005-0698-1. Google Scholar [17] J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. Google Scholar [18] G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moscow, 1946 (in Russian).Google Scholar [19] A. P. Veselov, Integrable discrete-time systems and difference operators, Funct. Anal. Appl., 22 (1988), 1-13. doi: 10.1007/bf01077598. Google Scholar
The discrete momentum locus $\mathfrak{u}^{(1)}_{\varepsilon}$ defined by $\ell_d^{(1, \varepsilon)}$ and the plane $\mathfrak{d}^*$ immersed in $\mathfrak{so}(3)^*=\mathbb{R}^3$. Although it cannot be appreciated from the figure, the surface is tangent to the plane at the origin
The discrete momentum locus $\mathfrak{u}^{(\infty)}_{\varepsilon}$ defined by $\ell_d^{(\infty, \varepsilon)}$ and the plane $\mathfrak{d}^*$ immersed in $\mathfrak{so}(3)^*=\mathbb{R}^3$. They intersect along the red curve, which self-intersects at the origin where $\mathfrak{u}^{(\infty)}_{\varepsilon}$ is tangent to $\mathfrak{d}^*$.
Comparison between the two discretisations for a generic inertia tensor. The graph on the left shows the approximation of $M_1$ and the evolution of the energy $E_c$ (inset). The graph on the right shows the approximation of $M_2$ and the evolution of the signed distance $\rho$ to the constraint subspace $\mathfrak{d}^*$ (inset)
Comparison between of the two discretisations for a special inertia tensor having $I_{11}=I_{22}$. The graph on the left shows the approximation of $M_1$ and the evolution of the energy $E_c$ (inset). The graph on the right shows the approximation of $M_2$ and the evolution of the signed distance $\rho$ to the constraint subspace $\mathfrak{d}^*$ (inset)
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