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

Luis C. garcía-Naranjo; Fernando Jiménez

doi:10.3934/dcds.2017182

Article Contents

2017, Volume 37, Issue 8: 4249-4275. Doi: 10.3934/dcds.2017182

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

Luis C. garcía-Naranjo^, and
Fernando Jiménez

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
^* Corresponding author: Luis C. García-Naranjo

Received: October 2016

Revised: March 2017

Published: May 2017

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

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Abstract

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.

Keywords:

Nonholonomic mechanics,
geometric integration,
discrete mechanics,
Lie groups and Lie algebras,
reduction of mechanical systems with symmetry.

Mathematics Subject Classification: Primary: 70F25, 37M99; Secondary: 65P10.

Citation:

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Figure 1. 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

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Figure 2. 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 $\mathfrak{d}^*$.

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Figure 3. 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)

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Figure 4. 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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