Various approaches to conservative and nonconservative nonholonomic systems

We propose a geometric setting for the Hamiltonian description of mechanical systems with a nonholonomic constraint, which may be used for constraints of general type (nonlinear in the velocities, and such that the constraint forces may not obey Chetaevʹs rule). Such constraints may be realized by servomechanisms; therefore, the corresponding mechanical system may be nonconservative. In that setting, the kinematic properties of the constraint are described by a submanifold of the tangent bundle, mapped, by Legendreʹs transformation, onto a submanifold (called the Hamiltonian constraint submanifold) of the phase space (i.e., of the cotangent bundle to the configuration manifold). The dynamical properties of the constraint are described by a vector subbundle of the tangent bundle to the phase space along the Hamiltonian constraint submanifold. In order to be able to deal with systems obtained by reduction by a symmetry group, we generalize that setting by using a Poisson structure on phase space, instead of the canonical symplectic structure of a cotangent bundle. The proposed geometric setting allows a very straightforward reduction procedure, which we compare with other reduction procedures, in particular with that of Bates and Śniatycki [5]. Possible generalizations for systems with controlled kinematic constraints are briefly indicated.