On some aspects of the discretization of the Suslov problem
For researchers in geometric numerical integration, this work offers theoretical insights into discretizing nonholonomic systems, though it is incremental.
The paper analyzes discretizations of the Suslov problem on SO(3), showing that consistency orders for unreduced and reduced setups are related nontrivially when using Cayley retractions. It provides conditions for constraint preservation and general consistency bounds.
In this paper we explore the discretization of Euler-Poincaré-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.