Based on the principle and viewpoint of the conservation of elastic energy of the mechanical structural system or elastic system that carries the structural body and its generalized characteristic features, the establishment of the understanding and analysis of the conservation of energy system to maintain the equilibrium of the sufficient conditions for the structural system of the total potential energy to have an extreme value (or stationary value). Since the total potential energy of the structural system is a general function, the problem of analyzing the stability of the equilibrium is mathematically reduced to the problem of solving the extremum of the general function, that is, a differential problem, with the help of the differential method, it is possible to establish the energy discriminant between the stable equilibrium of the (mechanical structure) system and the unstable equilibrium. In solving the stability problem, if the same linear assumptions are used in both the energy differentiation method and the static differential calculus, then both methods will lead to the same differential equations, so it can be said that the energy method and the static differential calculus are equivalent, however, for many practical problems, it is still easy to establish the differential equations, but it is not easy to solve the equations, and the problem can be solved if the energy method is used in the approximation of the problem, which means that the energy method is the same as the static differential calculus. That is to say, compared with the static method, the energy method is more attractive because it is particularly suitable for analyzing and explaining the principles of force transfer, transmission, change, disappearance and control, as well as approximate calculations. Therefore, the discussion herein focuses on introducing the energy method and utilizing the energy criterion to analytically investigate some of the basic concepts of nonlinear theory.