On some aspects of the discretization of the Suslov problem

TL;DR

This paper investigates the discretization of the Suslov problem on SO(3), analyzing the consistency and constraint-preserving properties of various integrators, and relating them to perturbations of the continuous system.

Contribution

It provides new insights into the relationship between unreduced and reduced discretizations using Cayley maps, and establishes conditions for constraint preservation in variational integrators.

Findings

Consistency order relations between reduced and unreduced setups.

Conditions for discrete flow to preserve constraints.

Discretizations can be viewed as exact flows of perturbed systems.

Abstract

In this paper we explore the discretization of Euler-Poincar\'e-Suslov equations on $SO(3)$, i.e. of the Suslov problem. We show that the consistency order corresponding to the unreduced and reduced setups, when the discrete reconstruction equation is given by a Cayley retraction map, are related to each other in a nontrivial way. We give precise conditions under which general and variational integrators generate a discrete flow preserving the constraint distribution. We establish general consistency bounds and illustrate the performance of several discretizations by some plots. Moreover, along the lines of [14] we show that any constraints-preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system.