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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 establish some general stability results for a large class of dynamical systems (which are arrived at via a Lagrange formulation). We then 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. The applicability of the present results is demonstrated by means of a specific example (a 162-bus, 17-generator model of the power network of the State of Iowa).