  We explore a particular approach to the analysis of dynamical and geometrical
properties of autonomous, Pfaffian non-holonomic systems in classical
mechanics. The method is based on the construction of a certain auxiliary
constrained Hamiltonian system, which comprises the non-holonomic mechanical
system as a dynamical subsystem on an invariant manifold. The embedding system
possesses a completely natural structure in the context of symplectic geometry,
and using it in order to understand properties of the subsystem has compelling
advantages. We discuss generic geometric and topological properties of the
critical sets of both embedding and physical system, using Conley-Zehnder
theory and by relating the Morse-Witten complexes of the 'free' and constrained
system to one another. Furthermore, we give a qualitative discussion of the
stability of motion in the vicinity of the critical set. We point out key
relations to sub-Riemannian geometry, and a potential computational
application.
