Symmetry and conserved quantity can simplify the dynamic problem and further obtain the exact solution of the mechanical system, which is more conducive to the study of dynamic behavior. Compared with the integer order model, the fractional model can describe the dynamic process of complex systems. Therefore, it is indispensable to study the symmetry and conserved quantities under the fractional model. Firstly, two fractional singular systems are introduced. One system contains mixed integers and Caputo fractional derivatives, and the other system contains only Caputo fractional derivatives. Two fractional inherent constraints are given by two fractional singular systems, and the corresponding fractional constrained Hamilton equation is given. Then, based on the invariance of differential equation under infinitesimal transformation, the definition of Lie symmetry of fractional constrained Hamilton equation is given, and the corresponding determined equation, restriction equation and additional constraint equation are derived. Thirdly, the Lie symmetry theorems of two fractional constrained Hamiltonian systems are established and proved, and the Lie conserved quantities of the corresponding fractional constrained Hamiltonian systems are obtained. Under certain conditions, the results obtained in this paper can be reduced to Lie conserved quantities of integer order constrained Hamiltonian systems. Finally, two examples are given to illustrate the application of this result.
沈世磊,宋传静.分数阶奇异系统的Lie对称性与守恒量[J].动力学与控制学报,2023,21(9):1~10; Shen Shilei, Song Chuanjing. Lie Symmetries and Conserved Quantities of the Fractional Singular Systems[J]. Journal of Dynamics and Control,2023,21(9):1-10.复制