Reconciling Kozlov’s vakonomic method with the traditional non-holonomic method: solution of two benchmark problems

We study the dynamics of a class of mechanical systems subjected to non-holonomic constraints by employing a method termed “modified vakonomic method” (MVM), and developed by Llibre, Ramírez and Sadovskaia in 2014. In particular, we test the MVM for the “rolling coin” problem and a variant of the “non-holonomic skate” problem. For our purposes, we divide our work in two parts. For the first one, our point of departure is a paper published in this journal by Lemos in 2022, in which, for the “rolling coin” problem, Kozlov’s vakonomic method is shown to lead to inconsistencies with the so-called Lagrange–D’Alembert traditional non-holonomic method (TNHM). In this case, we prove that, if the MVM is used, the equivalence with the TNHM can be restored, and the two methods can be reconciled. In the second part, we formulate a thought experiment consisting of an electrically charged “non-holonomic skate” interacting with a magnetic field, and we examine its dynamics by means of the MVM. In this case, we point out the differences with the predictions of the TNHM, and we propose a reformulation of the MVM capable of retrieving the results obtained with the TNHM. Moreover, we give some insight into the main computational aspects related to the MVM for non-holonomic constraints linear in the generalized velocities