Non-Holonomic Stability Aspects of Piecewise-Holonomic Systems

We consider mechanical systems with intermittent contact that are smooth and holonomic except at the instants of transition. Overall such systems can be nonholonomic in that the accessible configuration space can have larger dimension than the instantaneous motions allowed by the constraints. The known examples of such mechanical systems are also dissipative. By virtue of their non-holonomy and of their dissipation such systems are not Hamiltonian. Thus there is no reason to expect them to adhere to the Hamiltonian property that exponential stability of steady motions is impossible. Since non-holonomy and energy dissipation are simultaneously present in these systems, it is usually not clear whether their sometimes-observed exponential stability should be attributed solely to dissipation, to non-holonomy, or to both. However, it is shown here on the basis of one simple example, that the observed exponential stability of such systems can follow solely from the non-holonomic nature of intermittent contact and not from dissipation. In particular, it is shown that a discrete sister model of the Chaplygin sleigh, a rigid body on the plane constrained by one skate, inherits the stability eigenvalues of the smooth system even as the dissipation tends to zero. Thus it seems that discrete non-holonomy can contribute to exponential stability of mechanical systems ∗on leave 1997-8 at Automation Technology Lab, PB 3000, 02015 TKK, Otakaari 5 A, Espoo, FINLAND