An nonstability criterion of equlibrium for nonholonomic system

Ranislav Bulatović

Let $\Pi(\mathbf q)=\Pi_k(\mathbf q)+\Pi_{k+1}(\mathbf q)+\dots$, $\Pi_k(\mathbf q)\geq2$ and $A(\mathbf q)=A_0+A_s(\mathbf q)+\dots$, $s>1$, be McLaurin series of analytic potential and vector matrix of nonholonomic constraints. It can be proved that if there exist unit vector $\mathbf e\in R^n\{\mathbf q\}$ for which conditions $A^T_0e=0$, $\Pi_k(\mathbf e)=0$ and $\Pi_{k+1}(\mathbf e)<0$ are satisfied, then the equilibrium $\mathbf q=\bar{\mathbf q}=0$ is nonstable.