A New Method of Fractional Dynamics, I.E., Fractional Generalized Hamilton Method with Additional Terms, and its Applications to Physics

How to construct the fractional dynamical model is one of the most basic problems in fractional dynamics. However, for a long time, a large number of fractional differential equations of motion have been written directly by hand based on integer-order differential equations of motion. In this paper, we establish a new kind of fractional generalized Hamilton equation, and propose a new method to construct the fractional dynamical model, i.e. the fractional generalized Hamilton method with additional terms. As applications of the new method, we construct a series of new fractional dynamical models, which include a family of fractional Micro-electromechanical system (MEMS) with time-varying capacitance, a family of fractional Fokker-Planck model, two kinds of fractional Euler dynamical models of rigid body rotating around a fixed-point, three different kinds of fractional Van der Pol oscillator models and seven different kinds of fractional Duffing oscillator models. The successful construction of these models verifies the effectiveness and actual application value of the fractional generalized Hamilton method with additional terms. The work in this paper is of fundamental significance to the study of fractional dynamics.