Symmetry Theories Of Nonholonomic Systems With The Fractional Derivatives

Fractional calculus is a branch of calculus dealing with the generalization of the derivativeto a real order. Recently, fractional differentiation and fractional derivatives are now recognizedas vital mathematical tools to deal with the model of complex systems. Symmetries are theinvariance of the dynamics systems under the infinitesimal transformations, the concept ofsymmetries of mechanical systems plays an important role in mathematics and physics. In thispaper, we main study the symmetry theories and its inverse problems of nonholonomic systemswith the fractional derivatives.At first, we study the fractional Noether’s theorems and its inverse problems ofnonholonomic Lagrange systems. We introduce the two infinitesimal group of transformationscontain without the time variable and the general transformations of time-reparametrization. Thequasi-invariance of fractional nonholonomic Lagrange systems under the two infinitesimaltransformations are established respectively. Furthermore the fractional Noether’s theorems andthe formulism of fractional Noether’s conservation laws are acquired. We finally discuss thefractional inverse problems of Noether’s symmetry of nonhonomic Lagrange systems under theinfinitesimal transformations with the time variable.Secondly, we study the fractional Lie symmetries and its inverse problems of thenonholonomic Hamilton systems. The fractional equations of motion of systems are establishedby introducing the fractional generalized momentum. Then, determining equations, limitingequations and additional restricting equations, which are based on the invariance of the fractionalmotion equations, constraint equations and virtual displacement restrictive conditions of thesystems under the infinitesimal transformations of Lie symmetries, are acquired for the systems.Furthermore, definitions and theorems of the Lie symmetry, weak Lie symmetry and strong Liesymmetry are presented. Finally, we study the fractional inverse problems of Lie symmetry ofthe nonhonomic Hamilton systems.Finally, we establish the fractional Hojman conservation laws, which as a direct result ofthe invariance of the Lie symmetry of fractional nonholonmic Lagrange systems under a specialfractional infinitesimal transformation. Then we put forward the corresponding fractional Liesymmetry determining equations, limiting equations and the additional restricting equating.Moreover, we prevent fractional Lie symmetry theorem as a direct result of the Lie symmetry offractional nonholonmic Lagrange systems and the formulism of fractional Hojman conservationlaws.The contributions of this paper are listed below: (1) This paper adopts the methods of Lie group analysis, studying the symmetries andconservation laws of nonholonomic systems with the fractional derivatives.(2) Based on the Riemann-Liouville fractional theories, we studying the inverse problemsof symmetry of nonholonmic systems.(3) This paper presents a new symmetry method of solving the practical problems inengineering; providing the theory basics for acquiring the symmetries of the system with theknown first integrals.