This paper proposes to analyze the motion stability of synchronous generator-based power systems using a Lagrangian model derived in the configuration space of generalized position and speed. A Lagrangian model of synchronous generators is derived based on Lagrangian mechanics. The generalized potential energy of inductors and the generalized kinetic energy of capacitors are defined. The mechanical and electrical dynamics can be modelled in a unified manner by constructing a Lagrangian function. Taking the first benchmark model of sub-synchronous oscillation as an example

a Lagragian model is constructed

and a numerical solution of the model is obtained to validate the accuracy and effectiveness of the model. Compared with the traditional EMTP model in PSCAD

the obtained Lagrangian model is able to accurately describe the electromagnetic transient process of the system. Moreover

the Lagrangian model is analytical

which enables the analysis of the motion stability of the system using Lyapunov's motion stability theory. The Lagrangian model can not only be used for discussing the power angle stability but also for analyzing the stability of node voltages and system frequency. It provides the feasibility for studying the unified stability of power systems.