In this paper we consider nonholonomic control systems on Riemannian manifolds. Such systems evolve on subbundles of tangent bundles, defined by the nonholonomic constraints. This paper promotes the view of such systems as the restriction to the nonholonomic subbundle of "Newton law"-type problems on the entire tangent bundle, defined by, in general, non-Riemannian connections. These connections should be related to specific geometric properties of the nonholonomic system. We introduce a particular class of connections and demonstrate the richness of the class through four examples-the rolling ball, the constrained particle, the rolling penny, and the generalized rolling ball. This class of connections is strongly related to questions of integrability of the original nonholonomic system. This, in turn, provides additional insight into the relation between nonholonomic control systems formulated as kinematic equations and those that are formulated as the full dynamic equations.