On nonholonomic and vakonomic dynamics of mechanical systems with non integrable constraints

The classical nonholonomic equations for a mechanical system subject to linear nonintegrable constraints are presented in Section 2. These are displayed in a geodesic form by the use of a suitable covariant derivative (due to Synge) in Section 3. We then express the Frobenius integrability of the constraint distribution by means of a zero torsion condition for the above Synge connection. Section 4 provides a self-contained derivation from a nonholonomic variational problem of the equations of motion for nonholonomic systems. These equations, which are nonequivalent to the previous ones, were first developed by Arnold and Kozlov and called vakonomic (yak) equations. Sections 5 and 6 are concerned with a geometrical interpretation of the terms occurring in the right-hand side of the vak equations. Under quite general assumptions, these latter can be described in terms of the curvature of an Ehresmann (local) connection whose horizontal subspace is precisely the constraint distribution. Furthermore, by introducing a suitable tie group action on the configuration manifold, the local Ehresmann connection can be made into a global one which coincides with the mechanical connection of Smale-Marsden. Section 7 gives a motivation, in terms of Hopf-Rinow and Ambrose-Singer theorems, for the nonclassical requirement of the assignment of the reaction forces' values in the initial kinematical state in vakonomic mechanics. Section 8 develops a fundamental approach to the description of holonomic, i.e. geometrical constraints. We describe the reaction forces by using the Poincare dual (a class of closed I-forms) of the orientable constraint submanifold. As an instance of the construction developed in Sections 5 and 6, we consider in Section 9 the disk rolling without sliding on the plane.