State equilibrium stability of conservative systems in one particular case

Ranislav Bulatović

Conservative holonomic system, whose potential energy $P\in C^s$, $s\geq2$, in normal coordinates is: $P(\mathbf q)=\frac12\langle C(\mathbf q)\mathbf x,\mathbf x\rangle+P_k(\mathbf q)+W(\mathbf q)$, where $\mathbf q=(\mathbf x,\mathbf y)$, $\mathbf x\in\mathbb R^m$, $\mathbf y\in\mathbb R^n$, $C(\mathbf 0)=\operatorname{diag}(\omega^2_1,\dots,\omega^2_m)$, $P_k(\mathbf q)$ homogeneous polynomial of degre $k$, $k<s$ and $W(\mathbf q)=O(|\mathbf q|^{k+1})$ is studied. In a case when function $P_k(\mathbf 0,\mathbf y)$ has not minimum (has strict minimum) in $\mathbf y=\mathbf 0$, stability (instability) of the state of equilibrium $\mathbf q=\dot{\mathbf q}=\mathbf 0$ is proved. This statement, in a case when $P_k(\mathbf 0,\mathbf y)\geq0$, is completed with one additional instability criterion.