According to the Kirchhoff kinetic analogy method, the elastic deformation of a thin elastic rod with mutation of either curvaturetwisting vector or strain vector at section center was analogized as the striking motion. Based on the Cosserat and Kirchhoff thin elastic rod model, respectively, the Lagrange equations of the elastic rod with mutation of curvaturetwisting vector or section center strain vector were derived. It was found that the derived equations have the same forms as the Lagrange equation of striking motion. Finally, the effects of the mutation of curvaturetwisting vector or strain vector at section center on the smoothness of the deflection curve were analyzed. This study provides an analytical mechanics method for the equilibrium analysis of the elastic rod with mutation of curvaturetwisting vector or strain vector at section center.