Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data

Abstract

In this paper, we discuss the necessary and sufficient conditions for the equivalence of the dynamics of nonholonomic mechanics and variational nonholonomic (vakonomic) dynamics for certain initial conditions. We derive a priori results for identifying equivalence and, specializing to Abelian Chaplygin systems, prove that equivalence results if and only if the constrained nonholonomic equations are Lagrangian. We eliminate the need to solve the variational nonholonomic problem when checking equivalence by obtaining explicit formulae for the system's multipliers, and then derive conditions under which the multiplier free Lagrangian gives equivalence of the dynamics. We consider nonholonomic systems possessing invariant measures, showing when equivalence and Hamiltonization are the same. We also derive conditions under which measure-preserving systems exhibit equivalence. We apply the results to many of the known nonholonomic systems.