De Marco LINEARIZATION OF NONHOLONOMIC SYSTEMS AT EQUILIBRIUM POINTS

Linearization of nonholonomic systems at equilibrium positions has been plagued from the very beginning by mistakes and misunderstandings. Bottema [3] corrected Whittaker’s originary procedure [8]; and Naimark and Fufaev [7] developed Bottema’s idea, deriving a scheme which is the currently accepted one. Something nagging however remains about the derivation of the characteristic equation for a nonholonomic system: the equations of such a system, with the undetermined multipliers thrown in, do not form a system of ordinary differential equations, in spite of what is said in [7]; and the multipliers are not independent variables. The linearization being essentially an approximation, one can say that it is validated experimentally by the results it gives; but a satisfactory approach should be convincing also from the theoretical viewpoint. Here we try to do that by making use of the well–known fact that the equation of a nonholonomic system may be put in normal form, and the system then is equivalent to an holonomic system, in which only the initial conditions in phase space which satisfy the constraint are allowed. This fact has recently been used (see [9]) from Gaetano Zampieri, whom I thank for pointing out the question, and for many helpful discussions. The linearization of the equivalent holonomic system is a well understood procedure; we prove that the characteristic polynomial so obtained factors as ds2mβ(s), where β(s) is the Bottema polynomial, and d is a nonzero constant. Our approach is more complicated, but gives more insight into the question of stability, clarifies the insurgence of vanishing roots and makes possible the discovery of facts not apparent from the easier but more formal usual treatment. In particular, some instability results (see section 6) are consequence of this approach; we also discover that the characteristic equation has at least 2m vanishing roots, independently on any manifold structure of the set of equilibria. Equilibria of a nonholonomic system form, in the generic case, a manifold; but this is not always the case, as we show in section 7; and nonholonomic systems can have isolated equilibria (section 8). Finally, it is well–known that Whittaker’s procedure is acceptable for equilibria of the nonholonomic system which are also equilibria of the unconstrained system; some authors maintain that these are the only acceptable equilibria (see [4]); now, a skier on a slope can remain at rest when his skis are orthogonal to the slope’s gradient, even if the component of the gravity tangent to the slope is nonzero; this is obvious, and should end any further speculation on this matter.