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

Geometric integrators for nonholonomic systems were introduced by Cort\'es and Mart\'inez in [Nonholonomic integrators, Nonlinearity, 14, 2001] 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. Cort\'es and Mart\'inez 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.