The Motion Equations And Symmetry Theories Of Fractional Hamilton System

Fractional order calculus can be considered as a development of the integer order, which is a branch of the differential and integral calculus. In the field of modern engineering technology,many practical problems need to be described by fractional order models, and it becomes more and more important for the study of fractional Hamilton dynamics. In this paper, we have studied the Noether symmetry theories, non-Noether symmetry theories and Lie symmetry theories of fractional Hamilton systems, and proposed a new symmetry method for fractional problems. This paper studies the following aspects:Firstly, we establish the fractional motion equations with combined Caputo derivatives. The exchange relationship between the contemporaneous variation and the combined Caputo derivatives are given. Then, the fractional Hamilton principle with combined Caputo derivative is obtained. Furthermore, the fractional Lagrange equations and the fractional Hamilton’s canonical equations are obtained based on the fractional Hamilton principle.Secondly, we study the fractional cyclic integrals and Routh equations by the combined Caputo derivative. We put forward the fractional cyclic integrals based on the fractional Lagrange equations, and the associated Routh equations of the system are presented.Thirdly, the fractional motion equations and the fractional Poisson theorem of fractional Hamilton systems are set up by the fractional gene. We introduce a fractional gene and a fractional increment, and the fractional calculus with the fractional gene is presented. Then, the fractional Hamilton principle and fractional Hamilton’s canonical equations are given under this fractional calculus. We further study the fractional Poisson theorems by the fractional gene.Fourthly, the fractional Noether symmetry theories of Hamilton systems are presented by the conformable fractional derivative. The fractional Hamilton principle and fractional Hamilton’s canonical equations are given by the conformable fractional derivative. The fractional Noether symmetry transformation and quasi-Noether symmetry transformation are obtained. Then, we further study the Noether symmetry based on the fractional Noether symmetry transformation,and the corresponding Noether conserved quantities are acquired.Fifthly, the fractional non-Noether symmetry theories of Hamilton systems are given by the conformable fractional derivative. The fractional determining equations with conformable fractional derivatives are obtained according to the invariance of the differential equation under infinitesimal transformations. The non-Noether theorem with conformable fractional derivatives is presented based on the fractional determining equations, and the corresponding non-Noether conserved quantities are obtained.Sixly, we put forward the fractional Lie symmetry theories of Hamilton systems by the conformable fractional derivative. We study the fractional structural equation by the conformable fractional derivative. Then, the Lie symmetry theorem and the associated conserved quantities are given.