This paper proposes a method for transient stability analysis for multi-machine electric power systems. Different from existing methods, a minimization problem with boundary values is formulated for obtaining critical conditions for transient stability, where a new numerical integration method is developed by modifying the trapezoidal formula to solve effectively the boundary value problem. The proposed method is to compute directly a trajectory on the stability boundary, which is referred to as critical trajectory in this paper. The critical trajectory to be obtained is the trajectory that starts from a point on a fault-on trajectory and reaches a critical point such as an Unstable Equilibrium Point (UEP), or more exactly, Controlling UEP (CUEP). The solution of the minimization problem provides the critical trajectory and the exact Critical Clearing Time (CCT) without major approximation.