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In recent studies, it has been verified heuristically and experimentally (via simulations) that instability in power systems due to a fault occurs when one machine or a group of machines, called the critical group, loses synchronism with the remaining machines. Using energy functions associated with a critical group (rather than system-wide energy functions), transient stability results which are less conservative than other existing results, have recently been obtained. The existence and identity of a critical group is ascertained in these studies by off-line simulations. In the present paper, we first establish some general stability results for a large class of dynamical systems (which are arrived at via a Lagrange formulation). We then show how multi-machine power systems with nonuniform damping are special cases of this class of dynamical systems. Next, we show that our stability results can be used to establish analytically the existence and the identity of the critical group of machines in a power system due to a given fault. Furthermore, we also show that our stability results can be used to obtain an estimate of the domain of attraction of an asymptotically stable equilibrium of a power system. The results presented herein can potentially be used on-line to determine which machines belong to a critical group, and to use this information for corrective action (e.g., shedding of the critical generators or fast valving for these generators.) As such, our results have potential applications in fault diagnosis and security assessment of certain classes of dynamical systems in general and of power systems in particular. The applicability of the present results is explored by means of a specific example (a 162-bus, 17-generator model of the power network of the State of Iowa). Certain limitations of the present results are recognized and avenues for further research are identified.