New Methods For Constructing Fractional Dynamics Model:the Fractional Generalized Hamilton Method With Additional Terms

Fractional dynamics has become a crucial topic in the field of science and engineering.How to construct the model of fractional dynamics is the most basic problem in fractional dynamics.However,for a long time,a large number of differential equations of motion for fractional dynamical systems have been written directly by hand! In order to solve this problem,Luo Shaokai proposed five analytical mechanics methods for fractional dynamics,fractional generalized Hamilton method is one of them,please refer to the review "Analytical mechanics methods and applications for fractional dynamics",Journal of Dynamics and Control(2019)(5).However,for complex dynamical systems,especially those that cannot be fully fractional Hamiltonized,it is invalid to construct fractional dynamical models by using fractional generalized Hamilton method.Therefore,the basic theory and method of fractional generalized Hamilton system with additional terms are proposed.This paper has carried out in-depth and systematic research work around this topic,and verified the effectiveness and practical application value of the new method.In Sect.1 explains briefly the history and present situation of fractional dynamics and the research progress of fractional generalized Hamilton mechanics,and presents an crucial problem to be solved in this paper: the basic theory and method of fractional generalized Hamilton systems with additional terms.In Sect.2,firstly,the definitions of four different fractional derivatives and their main properties are introduced.Secondly,the fractional generalized Hamilton equation under the definition of Riesz-Riemann-Liouville derivative is given.Then,the degradation conditions for the fractional generalized Hamilton equation to be reduced to the fractional Hamilton equation are given.Furthermore,the fractional generalized Hamilton method for constructing the fractional dynamics model is given.Finally,as an application of the fractional generalized Hamilton method,the fractional general relativistic Buchduhl model,the fractional Lotka biochemical oscillator model,the fractional Lorentz-Dirac model,the fractional Whittaker model,the fractional Henon-Heiles model and the fractional relativistic Yamaleev oscillator model are constructed.In Sect.3,the basic theory and method of fractional generalized Hamilton system dynamics with additional terms are given.Firstly,three new fractional generalized Hamiltonian equations with additional terms are given for dynamic systems which can be fully or partially fractional Hamiltonized.Then,fractional generalized Hamilton method with additional terms is proposed to construct the fractional dynamics model.Further,the generalized fractional Hamilton equation with additional terms is degenerated,transformed or extended to obtain the fractional Hamilton equation with additional terms,the fractional Lagrange equation with additional terms,the fractional Birkhoff equation with additional terms,and the fractional Nambu equation with additional terms,respectively.The fractional Hamilton method,the fractional Lagrange method,the fractional Birkhoff method and the fractional Nambu method are proposed to construct the fractional dynamics model.In Sect.4,as new method's applications,four families of new fractional dynamics model are constructed based on the fractional generalized Hamilton method with additional terms,including: a family of fractional MEMS model with time-varying capacitances,a family of fractional three-particle Toda lattice system model,a family of fractional Emden-Fowler system model and a family of the fractional Fokker-Planck system model.In Sect.5,as new method's applications,based on the fractional generalized Hamiltonian method with additional terms,fractional Euler-Poinsot model of rigid body that not subjected to external moments of force is constructed,fractional Euler model of rigid body that subjected to external moments of force is also constructed.In Sect.6,as new method's applications,three kinds of fractional Van der Pol oscillator models are constructed based on the fractional generalized Hamilton method with additional terms,including: fractional Van der Pol model with nonlinear damping force,fractional Van der Pol model subjected to external force,fractional Duffing-Van der Pol oscillator model.In Sect.7,as new method's applications,seven kinds of fractional Duffing oscillator models are constructed based on the fractional generalized Hamilton method with additional terms,including: fractional Duffing oscillator model with nonlinear elastic restoring force,fractional Duffing oscillator model with nonlinear elastic restoring force and damping force,fractional Duffing oscillator model with nonlinear elastic restoring force,damping force and external exciting force,fractional Duffing oscillator model with nonlinear elastic restoring force,damping force,external exciting force and Gaussian white noise,fractional Rayleigh-Duffing oscillator model,fractional Duffing-like oscillator model.Based on a series of fractional dynamics models constructed by us,the intrinsic properties and dynamics behaviors of different fractional systems can be further explored by analytical or numerical methods.Section 8 summarizes the innovative work in this paper,and gives some suggestions on the basic theory and method of fractional generalized Hamilton system with additional terms.