Abstract: According to Kirchhoff’s kinetic analogy, the concepts and approaches of dynamics can be applied to mechanics of elastic rods. The Cosserat model of elastic rods considers factors neglected by the Kirchhoff model, such as the cross-sectional shear deformation, the tensile deformation of rod centerline and loadings, and therefore is more suitable to the modeling of thin-long beams with large deformation. In this study, the Cosserat model is applied to establish the exact dynamic equations of a thin-long curve beam with circular cross sections and an arbitrary shape of centerline. The attitude angles of rod cross sections and deflections of the centerline are selected as unknown variables. The Timoshenko beam in elastic medium represents the special case with beam of a straight centerline and small deformation. The Lyapunov’s stability theory is applied to analyze the equilibrium status of the beam when temporal variables are replaced by spatial variables. The stability of a Timoshenko beam in elastic medium subject to axial compression is analyzed to identify the difference and relationship between Lypunov’s stability and the traditional Euler’s buckling theory. The Euler’s load is obtained by using the perturbed centerline of the beam satisfying constraint conditions on beam ends when the Lyapunov’s stability conditions are satisfied. It is proven that the Lyapunov’s and Euler’s stability conditions in the spatial domain are the necessary conditions of Lyapunov’s stability in the time domain in the sense of the first-order approximation.