Research On Basic Theory And Method Of Conformal Invariance Of Fractional Dynamical Systems

The symmetry and conformal invariance of dynamical system is a general and important property in mechanics,physics,mathematics and engineering science,and it has various applications for the research of actual dynamical model.Since 1996,international mathematicians have established the fractional Lagrangian equation,fractional Hamiltonian equation and fractional dynamical equations of nonholonomic system consecutively.In recent years,Luo Shaokai has proposed and led the graduate students establishing new fractional Lagrangian mechanics,fractional Hamiltonian mechanics,fractional generalized Hamiltonian mechanics,fractional Birkhoff mechanics and fractional Nambu mechanics with complete dynamic information,and constructed their theoretical framework respectively,revealed the internal structure and dynamical behaviors of a dynamical systems which mainly include these systems' gradient representation,algebraic structure,Poisson conservation law,variational equation,integral Invariants,motion stability etc.But the conformal invariance of dynamical systems still need to be further researched.To solve this problem,based on the definition of fractional derivative of Riesz,this dissertation will study the conformal invariance of fractional Lagrange system,fractional Hamiltonian system,fractional generalized Hamiltonian system,fractional Nambu system and fractional Birkhoffian system with complete dynamic information.The conformal invariant method of Mei symmetry and the conformal invariant method of Lie symmetry are given respectively to find the conserved quantities of fractional dynamics and the conserved quantities of fractional dynamical systems are obtained.The applications in actual fractional dynamics model are also studied.Thus,we can find new methods for solving related scientific and engineering problems by enriching and developing the theory of fractional dynamics and the conformal invariance of dynamical systems.Section 1 explains briefly the history and status of conformal invariance of dynamical systems and fractional dynamics,and presents the problems to be solved in this paper.In Sect.2,firstly,we introduce the definitions and the properties of the Riemann–Liouville,the Caputo,the Riesz–Riemann–Liouville and Riesz–Caputo fractional derivatives respectively.Then,based on the definition of Riesz–Riemann–Liouville fractional derivative,we give the fractional Lagrangian equation,fractional Hamilton equation,fractional generalized Hamilton equation,fractional Birkhoff equation and fractional Nambu equation;and present the fractional Lagrangian method,fractional Hamilton method,fractional generalized Hamilton method,fractional Birkhoff method and fractional Nambu method for constructing the fractional dynamical models.In Sect.3,based on Lagrange representation of fractional dynamics system,we study the conformal invariance,conformal invariance and Mei symmetry,conformal invariance and Lie symmetry of fractional Lagrange system.The corresponding judgment theorems are given respectively and the corresponding conserved quantities are obtained.As applications,we construct actual fractional Kepler model and fractional Hénon-Heiles model and explore these models' conformal invariance and conserved quantities respectively.In Sect.4,based on Hamilton representation of fractional dynamics system,we study the conformal invariance,conformal invariance and Mei symmetry,conformal invariance and Lie symmetry of fractional Hamiltonian system.The corresponding judgment theorems are given respectively and the corresponding conserved quantities are obtained.As applications,we construct the fractional Hénon-Heiles and fractional Emden models,and explore these models' conformal invariance and conserved quantities respectively.In Sect.5,based on generalized Hamilton representation of fractional dynamics system,we study the conformal invariance,conformal invariance and Mei symmetry,conformal invariance and Lie symmetry of fractional generalized Hamiltonian system.The corresponding judgment theorems are given respectively and the correspondingconserved quantities are obtained.As applications,we construct fractional generalized relativistic Buchduhl model,fractional order Duffing oscillator model and fractional Whittaker model,and explore these models' conformal invariance and conserved quantities respectively.In Sect.6,based on the Nambu representation of fractional dynamical system,we study the conformal invariance,conformal invariance and Mei symmetry,conformal invariance and Lie symmetry of fractional Nambu system.The corresponding judgment theorems are given respectively and the corresponding conserved quantities are obtained.As applications,we construct fractional relativistic Yamaleev oscillator model and fractional Duffing oscillator model,and explore these models' conformal invariance and conserved quantities respectively.In Sect.7,based on Birkhoff representation of fractional dynamics system,we study the conformal invariance,conformal invariance and Mei symmetry,conformal invariance and Lie symmetry of fractional Birkhoffian systems.The corresponding judgment theorems are given respectively and the corresponding conserved quantities are obtained.As applications,we construct fractional Lotka biochemical oscillator model,fractional Hojman-Urrutia model and fractional Lorentz-Dirac model,and explore these models' conformal invariance and conserved quantities respectively.Section 8 concludes the research results of this paper,and gives some suggestions for further research on conformal invariance of fractional dynamical systems.