Conformal Invariance And Conserved Quantity Of Fractional Constrained Mechanical Systems

This paper studies the conformal invariance and conserved quantities of fractional constrained mechanical systems.The conformal invariance theory is studied in fractional Lagrangain systems,fractional nonholonomic Lagrangain systems,fractional nonconservative systems in phase space,and fractional Birkhoffian systems.Starting from the fractional calculus theory,the relationships between the fractional conformal invariance and the Lie symmetry of the system is studied,and the expression of the corresponding system’s conformal factor is obtained.It can study the conformal invariance of Lie symmetry in different fractional dynamical systems,and the corresponding conserved quantity is established.It is of great theoretical and practical value to study the conformal invariance of mechanical systems under fractional models.It will break through the traditional mechanical system’s conformal invariance and conserved quantity theory research into the category of integer mechanical systems,enriching and developing.The symmetry and conserved quantity theory of fractional mechanical systems provide a new theoretical basis for further study of the intrinsic properties and potential laws of fractional dynamic systems.The research in this paper mainly includes the following aspects:First,based on the Riemann-Liouville fractional derivative,the conformal invariance and conserved quantity of the fractional Lagrangain system are studied.The differential equations of the fractional Lagrangain system are established,and the definition of conformal invariance of the fractional Lagrange system is given.The relationships between the conformal invariance of the fractional Lagrangain system and the Lie symmetry is given,and the conformal factor is obtained.The expression and the existence of the Noether conserved quantity of the conformal invariance under the Lie symmetry of the fractional Lagrangain system are given.Secondly,based on the Riemann-Liouville fractional derivative,the conformal invariance and conserved quantity of fractional nonholonomic Lagrangain systems are studied.The differential equations of fractional nonholonomic Lagrangian systems are established.The definition of conformal invariance of fractional order nonholonomic Lagrangain systems is given.The agreement between the conformal invariance and Lie symmetry of fractional order nonholonomic Lagrange systems is given.The relation is obtained,and the expression of the conformal factor is obtained.The condition and form of the existence of the Noether conserved quantity of the conformal invariance under the Lie symmetry of the fractional order nonholonomic Lagrangain system are given.Thirdly,based on the Caputo fractional derivative,the conformal invariance and conserved quantity of phase space nonconservative systems with fractional order are studied.The Hamiltonian regular equations of phase space nonconservative mechanical systems are established.The definition of conformal invariance of phase space nonconservative mechanical systems is given.The conformal invariance of phase space nonconservative mechanical systems is given.The relationships between Lie symmetry is obtained,and the expression of the conformal factor is obtained.The condition and form ofthe existence of the Noether conserved quantity under the Lie symmetry of the phase space nonconservative mechanical system are given.Fourthly,based on the Riemann-Liouville fractional derivative,the conformal invariance and conserved quantity of the fractional Birkhoffian system are studied.The differential equations of the fractional Birkhoffian system are established,and the definition of the conformal invariance of the fractional Birkhoffian system is given.The relationships between the conformal invariance and the Lie symmetry of the fractional Birkhoffian system is given.The condition and form of the existence of the conformal invariance Noether type under the Lie symmetry of the order Birkhoffian system.