On the Hamiltonian Formulation of Nonholonomic Mechanical Systems

The theory of mechanical systems with nonholonomic constraints has a long history in classical mechanics; see e.g. the books by Neimark & Fufaev [14], Edelen [6], Rosenberg [16], Arnold [l] and the references quoted in there. In this literature, nonholonomic mechanical systems are described within the variational framework by Euler-Lagrange equations with extra terms corresponding to the constraint forces. The present note is largely influenced by a recent paper of Bates & Sniatycki [4], see also Stanchenko [17], where it is shown that the dynamics of mechanical systems with nonholonomic constraints may be alternatively described within a Hamiltonian framework. However, the two-form with respect to which the Hamiltonian equations of motion (on a reduced state space, and without constraint forces) are defined is not necessarily closed, as may be demonstrated on simple examples. As a consequence, the resulting equations of motion, albeit of a Hamiltonian format, need not admit canonical coordinates and thus need not be transformable to the standard Hamiltonian equations: Gi = E, & = e, i = 1, . . . , n. In the present note we will use, instead of the notion of a (not necessarily closed) two-form, the dual object of a “Poisson” bracket not necessarily satisfying the Jacobi identity. We will show in a simple manner that the dynamics of mechanical systems with holonomic or nonholonomic constraints is Hamiltonian with respect to such a generalized bracket. An explicit expression for this bracket is provided. Furthermore, we will show that this