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 (non-linear in the velocities, and such that the constraint forces may not obey Chetaev's 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 Legendre's transformation, onto a submanifold (called the Hamiltonian constraint submanifold) of the phase space (i.e., of the cotangent bundle to the connguration 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 that of Bates and Sniatycki 5]. Possible generalizations for systems with controlled kinematic constraints are brieey indicated.