Asymptotic motions of mechanical systems with non-holonomic constraints

The motions of mechanical systems with non-holonomic constraints close to critical points of the potential are considered. The stability of the equilibrium positions was first treated by Whittaker /1/. A theorem is given which includes earlier results /2/ as a special case, and which enables asymptotic motions to be found for new classes of potentials. Sufficient conditions are found for the equilibrium to be unstable when not all the frequencies of small oscillations vanish. Similar studies were made in /3-6/ for systems without constraints. The hypothesis can be advanced that a critical point of the potential energy is an unstable equilibrium of a mechanical system with non-holonomic constraints (linear in the velocity) when zero is not a minimum of the function V*. Here, the origin is the equilibrium position in question, and the asterisk denotes contraction of the potential energy V to the subspace, orthogonal to all the constraints at zero. This hypothesis is proved below for the case when the MacLaurin expansion of V* is V* = V₂* + V_k* + v*_k+1 + ..., where V₂* + V_k* can take negative values infinitesimally close to zero (V_j* is a homogeneous form of degree j). This situation when V₂* ≥ 0 and V_k* ≥ 0 is not considered. Also, to determine the absence of a minimum, higher powers must be taken into account.