The Non-Holonomic Mechanics

Two approaches for the study of mechanical systems with non-holonomic constraints are presented: dʹAlembertian mechanics and variational (vakonomic) mechanics. The first one is equivalent to the dʹAlembert principle and the second comes from a variational principle. In order to develop the two approaches, dʹAlembertian and vakonomic trajectories are introduced. A generalized version of the classical Liouville theorem for the conservation of volume is proved in the context of dʹAlembertian mechanics. A characterization for the notions of regular and singular curves is presented. The flow corresponding to the regular vakonomic trajectories is Hamiltonian