This paper proposes a new formulation for transient stability analysis for electric power systems. Different from existing methods, a minimization problem is formulated for obtaining critical clearing time (CCT) for transient stability. The method is based on the computation of a trajectory on the stability boundary, which is referred to as critical trajectory in this paper. The critical trajectory is defined as the trajectory that starts from a point on a fault-on trajectory at CCT and reaches a critical point of losing synchronism. The new proposal includes a modified trapezoidal formulation for numerical integration, the critical conditions for synchronism, and the unified minimization formulation. It will be demonstrated that the solution of the minimization problem successfully provides the exact CCT that agrees with the conventional numerical simulation method.