| The research of fractional dynamical system is a leading academic issue in the field of international science and engineering,causing the attention of scientists in various fields.However,solving a fractional differential equation is a fundamental and difficult problem!With the development of science since 1788,analytical mechanics scientists present a series of integration method to solve dynamical equations,for example,Poisson method,Jacobi last multiplier method,Lie symmetrical method,Mei symmetrical method of finding conserved quantities and so on.And these classical integration methods had been applied to solving integer order differential equations.In the last 20 years,international scientists had respectively established fractional Lagrange equation,fractional Hamilton equation and fractional nonholonomic system dynamical equation.Since 2010,the research group leading by Luo has established new fractional Lagrangian mechanics,holonomic fractional Hamiltonian mechanics,fractional generalized Hamiltonian mechanics,fractional Birkhoff mechanics and fractional Nambu mechanics,and then studied these systems’ gradient representation,algebraic structure,Poisson conservation law,variational equation,integral Invariants,motion stability etc.But the question is that based on analytical mechanics representation of fractional differential equation whether the classical integration methods of analytical mechanics can be used to solve the fractional differential equations?Based on the definition of the Riesz–Riemann–Liouville fractional derivative,using the fractional Lagrange representation,the fractional Hamiltonian representation,fractional generalized Hamiltonian representation,fractional Nambu representation and fractional Birkhoff representation of fractional differential equations to study the analytical mechanics method of fractional differential equations,mainly include fractional Jacobi last multiplier method,fractional Lie symmetrical method and fractional Mei symmetrical method of fractional differential equations,and research the applications of these three methods to the actual dynamical models.In theory to extend the theory of fractional dynamics and fractional differential equations;in method to present three approaches of solving actual dynamical models;in application to study the analytical mechanics methods of several typical actual fractional models.And these methods provide a reference for exploring the intrinsic properties and dynamic behavior of other actual models.These methods have important theoretical value and broad practical value in modern mathematics,mechanics,physics and engineering,which also enrich and develop the theory and method of fractional order dynamics and fractional differential equation.Section 1 explains briefly the history and status of the analytical mechanics and fractional dynamics,and presents the problems to be solved in this paper.In Sect.2,firstly,we respectively introduce the definitions and the primary properties of the Riemann–Liouville,the Riesz–Riemann–Liouville,the Caputo and Riesz–Caputo fractional derivatives.And then,based on the definition of Riesz–Riemann–Liouville fractional derivative,we give the fractional Lagrange representation,the fractional Hamiltonian representation,fractional generalized Hamiltonian representation,fractional Nambu representation and fractional Birkhoffian representation of fractional differential equations;and present the fractional Lagrangian method,fractional Hamilton method,fractional generalized Hamilton method,fractional Nambu method and fractional Birkhoffian method for constructing the fractional dynamical models.In Sect.3,we present a general method of finding conserved quantities,i.e.,fractional Jacobi last multiplier method.With the definition of Riesz–Riemann–Liouville fractional derivative,we study general fractional dynamical equations,construct its fractional Jacobi last multiplier and,respectively,give determining equation and three important properties of the multiplier.And then,we present fractional Jacobi last multiplier method,which includes three theorems of finding conserved quantities of fractional dynamical system.Furthermore,the fractional Jacobi last multiplier method is applied to the fractional Lagrange system,the fractional Hamiltonian system,the fractional generalized Hamiltonian system,the fractional Nambu system and the fractional Birkhoffian system,and five corresponding propositions are given.Also by using the fractional Jacobi last multiplier method,we respectively find the conserved quantities of a fractional relativistic Buchduhl model,a fractional Robbins–Lorenz model,a fractional Euler–Poinsot model and a fractional Duffing oscillator model.In Sect.4,we present a general method of finding conserved quantities,i.e.,fractional Lie symmetrical method.With the definition of Riesz–Riemann–Liouville fractional derivative,based on the fractional Lagrange representation,the fractional Hamiltonian representation,fractional generalized Hamiltonian representation,fractional Nambu representation and fractional Birkhoffian representation of fractional differential equations,we present a fractional Lie symmetrical method for finding conserved quantities of fractional systems,including construct a new kind of single-parameter fractional infinitesimal transformation of Lie group in ?-1 order space and,under this transformation,obtain the fractional Lie symmetrical determining equation and the theory for solving conserved quantities of the fractional systems.And using the analytical mechanics representation of fractional differential equations and fractional Lie symmetrical method to find the conserved quantities of actual dynamical systems,and we respectively find the conserved quantities of a fractional Hénon–Heiles model,a fractional Emden model,a fractional Lotka biochemical oscillator model,a fractional Duffing oscillator model and a four-dimensional fractional Birkhoffian model.Further,in fractional framework,we explore the relation between Lie symmetry and Jacobi last multiplier.In Sect.5,we present a general method of finding conserved quantities,i.e.,fractional Lie symmetrical method.With the definition of Riesz–Riemann–Liouville fractional derivative,based on the fractional Lagrange representation,the fractional Hamiltonian representation,fractional generalized Hamiltonian representation,fractional Nambu representation and fractional Birkhoffian representation of fractional differential equations,we present a fractional Mei symmetrical method for finding conserved quantities of fractional systems.Under the general single-parameter fractional infinitesimal transformation of Lie group,we respectively obtain the fractional Mei symmetrical determining equation and the theory for solving conserved quantities of the fractional systems in different representations.And using the analytical mechanics representation of fractional differential equations and fractional Mei symmetrical method to find the conserved quantities of actual dynamical systems,and we respectively find the conserved quantities of a fractional Kepler model,a fractional Hénon–Heiles model,a fractional relativistic Buchduhl model,a fractional relativistic Yamaleev oscillator model and a Hojman–Urrutia model.Section 6 concludes the major research results of this paper,and gives some suggestions for further research on the analytical mechanics methods of fractional differential equations. |
