A stability problem with respect to a part of variables of the zero equilibrium position is considered for nonlinear non-stationary systems of ordinary differential equations with the continuous right-hand side. As compared to known assumptions, more general assumptions are made on the initial values of variables non-controlled in the course of studying stability. In addition, a stability problem is considered with respect to a part of variables of the “partial” equilibrium position, with similar assumptions made for initial values of variables that do not define the given equilibrium position. Conditions of stability and asymptotic stability of this type are obtained within the method of Lyapunov functions and generalize a number of existing results. The results are applied to the stability problem with respect to a part of variables of equilibrium positions of nonlinear holonomic mechanical systems. The problem of unification (to a certain extent) of the process of studying partial stability problems of stationary and non-stationary systems is discussed.