A solution method for decomposing vector fields in Hamilton energy

Hamilton energy, which reflects the energy variation of systems, is one of the crucial instruments used to analyze the characteristics of dynamical systems. Here we propose a method to deduce Hamilton energy based on the existing systems. This derivation process consists of three steps: step 1, decomposing the vector field; step 2, solving the Hamilton energy function; and step 3, verifying uniqueness. In order to easily choose an appropriate decomposition method, we propose a classification criterion based on the form of system state variables, i.e., type-I vector fields that can be directly decomposed and type-II vector fields decomposed via exterior differentiation. Moreover, exterior differentiation is used to represent the curl of low-high dimension vector fields in the process of decomposition. Finally, we exemplify the Hamilton energy function of six classical systems and analyze the relationship between Hamilton energy and dynamic behavior. This solution provides a new approach for deducing the Hamilton energy function, especially in high-dimensional systems.