The stability of equilibrium positions in critical cases according to Liapunov

The determination of the stability characteristics of the equilibrium positions of dynamic systems is of great practical importance in technology. The present investigation is concerned with a scleronomic mechanical system with m degrees of freedom. The equations of motion can be written in local coordinates with respect to the steady solution as system of n (n equal to or less than 2m) ordinary autonomous differential equations of the first order. The occurrence of a critical stability case according to Liapunov or a degenerate bifurcation problem is considered. In the case of the degenerate stability problem, transformation into a nondegenerate, i.e., a stability problem solvable by a linear approach is required. It is found that for some considered problems the critical stability case according to Liapunov can be systematicaly solved without the employment of Liapunov functions.