Mechanical systems with nonholonomic constraints can be analyzed within a comprehensive geometric framework based on Hamiltonian mechanics. The dynamics of such systems are determined by constructing vector fields on constraint submanifolds, with particular emphasis on the underlying symplectic geometry. We extend this framework to accommodate non-smooth constraints, enabling the analysis of systems exhibiting discontinuities such as collisions and sliding phenomena. Two mechanical systems are studied in detail: a pendulum with a sliding mass, and an inverted pendulum with an oscillating suspension point. For the sliding mass pendulum, we establish stability properties of the vertical position and identify specific nonholonomic constraints that achieve asymptotic stabilization. For the inverted pendulum, we derive the Mathieu equation directly from the geometric formulation and recover the classical stability criterion that accounts for the counterintuitive stabilization of the upright position via rapid oscillations. This methodology provides a unified treatment of both smooth and non-smooth dynamics, offering insights into control strategies for mechanical systems and illustrating the power of geometric methods in analyzing complex dynamical behavior.