The Study Of Lie Symmetry And Conserved Quantity For Holonomic Mechanical Systems Based On Fractional Models

Noether's theorem is the study of the Hamilton action invariance under the infinitesimal transformations,and the corresponding symmetry is called Noether symmetry.Lutzky and others found the Noether symmetry can't include all the symmetries,which enables people to know the symmetry again.In 1979,Lutzky applied the invariance theory of differential equations of motion of Sophus Lie into the field of mechanical system,and give the concept of Lie symmetry,which extends the symmetry and invariants area of research.This paper will mainly study the Lie symmetry of holonomic mechanical systems based on the fractional models,and the corresponding conserved quantities will be obtained.First,the D'Alembert-Lagrange principle of the El-Nabulsi fractional models is deduced based on the fractional action-like variational problem which is expanded by the Riemann-Liouville integral,and the fractional Euler-Lagrange equations of the system are obtained.The determination equations of the system are established,and the definition and the criterion of the Lie symmetry of the Lagrange system based on El-Nabulsi fractional models are given,and the generalized Hojman theorem is put forward.At the same time,the existence condition and the form of the generalized Hojman conserved quantity are obtained.The generalized Noether theorem is established,the existence condition and the form of the Noether conserved quantity led by the fractional Lie symmetry are given.Second,the fractional Hamilton canonical equations of the system are established.The determination equations of the system are established,and the definition and the criterion of the Lie symmetry of the Hamilton system based on El-Nabulsi fractional models are given,and the generalized Hojman theorem is put forward.At the same time,the existence condition and the form of the generalized Hojman conserved quantity are obtained.The generalized Noether theorem is established,the existence condition and the form of the Noether conserved quantity led by the fractional Lie symmetry are given.Third,the fractional principle of D'Alembert-Lagrange is deduced based on the non-conservative Hamilton principle and the fractional Euler-Lagrange equations are established.The Lie symmetry under the general infinitesimal transformations is researched and its determination equations are established,and the definition and the criterion of the Lie symmetry of the non-conservative Lagrange system based on Agrawal fractional models are given.The existence condition and the form of the conservedquantity led by the fractional Lie symmetry are given.Fourth,the fractional Hamilton canonical equations are established based on the Hamilton principle for the non-conservative system.The determining equations under the infinitesimal transformations of a group are given,and the definition and the criterion of the Lie symmetry of the non-conservative Hamilton system based on Agrawal fractional models are established.The condition under which a Lie symmetry can lead to a new type of fractional conserved quantity is gained and the form of the conserved quantity is presented.In this paper,the Lie symmetry and conserved quantity of Lagrange system,Hamilton system based on El-Nabulsi fractional models are studied,and the Lie symmetry and conserved quantity of nonconservative Lagrange system,nonconservative Hamilton system based on Agrawal fractional models will be also studied.The integer Lie symmetry is a special case of the results of this paper which is general.