Integrating Factors And Conservation Laws For Fractional Mechanical System

In order to further study the conserved quantity for mechanical systems,in this paper, the method of integrating factors is extended to mechanical systems in the case of integer order(the mechanical system with non-Chetaev nonholonomic unilateral constraints,the generalized Birkhoffian system) and fractional case(the fractional nonholonomic system based on the fractional model extended periodic fractional integral, the Lagrange system and Hamilton system based on the Riemann-Liouville fractional model). The equations which are used to determine the integrating factors are presented by finding the necessary conditions for the existence of the conserved quantities of the corresponding system and establishing the relation between the conserved quantities and the integrating factors. Then, the conserved quantities of the corresponding system could be found.In the first part, the method of integrating factors is applied to mechanical systems with non-Chetaev nonholonomic unilateral constraints. First, the necessary conditions for the existence of the conserved quantity of nonholonomic mechanical systems are studied. Second, the relation between the conserved quantities and the integrating factors is established, then the equations which are used to determine the integrating factors can be presented. Finally, the conserved quantities of non-Chetaev nonholonomic mechanical systems with unilateral constraints be found. Also, an example is given for its application.In the second part, the method of integrating factors for Birkhoffian systems is extended to a generalized Birkhoffian system. The equations which are used to determine the integrating factors are presented by finding the necessary conditions for the existence of the conserved quantities of a generalized Birkhoffian system and establishing the relation between the conserved quantities and the integrating factors. Then, the conserved quantities of the generalized Birkhoffian system could be found. Also, an example is given for its application.In the third part, the method of integrating factors is applied to the fractional nonholonomic system based on the fractional model extended periodic fractional integral. First, the necessary conditions of the existence of the conserved quantities of the fractional Routh equations and the relation between the conserved quantities and the integrating factors are studied. Second, the fractional equations which are used to determine the integrating factors are presented. Finally, the conserved quantities of the fractional nonholonomic system based on the fractional model extended periodic fractional integral be found. Also, an example is given for its application.In the forth part, the method of integrating factors is applied to the fractional Lagrange system and Hamilton system which are based on the Riemann-Liouville fractional model. First, the necessary conditions for the existence of the conserved quantities for the fractional mechanical systems and the relation between the conserved quantities and the integrating factors are established. Second, the fractional equations which are used to determine the integrating factors are presented. Finally, the conserved quantities of the fractional mechanical systems based on the Riemann-Liouville fractional model be found. Also, examples are given for the application of the results.