Modeling of Constrained Systems

The equations of motion for a mechanical system with nonholonomic velocity constraints are derived using the assumption that the generalized force vector must at each instant annihilate the space of all possible virtual displacements. We show how this assumption is in fact a consequence of more elementary axioms, such as Newton's second law (F = ma) for each particle of the system, or the equations of balance of linear and angular momentum for rigid bodies. Our argument relies on a careful consideration of how the generalized force vector depends on the physical forces on the system. Even in problems where the generalized force vector does not annihilate the space of all possible virtual displacements (as in problems involving sliding friction) our methodology gives an e cient way for obtaining the the equations of motion in generalized coordinates.