We introduce a method which allows one to recover the nonholonomic equations of motion of certain systems by instead finding a Hamiltonian via Pontryagin¿s maximum principle on an enlarged phase space, and then restricting the resulting canonical Hamilton equations to an appropriate invariant submanifold of the enlarged phase space. We illustrate the method through several examples, and discuss its relationship to the enlarged phase space. We illustrate the method through several examples, and discuss its relationship to the integrability of the system, and its quantization integrability of the system, and its quantization.