Polynomial Integrability Of A Class Of Hamiltonian Systems

Hamiltonian system is an expression of the dynamic system described by Hamiltonian mechanics and is an important part of dynamical systems,which was originally proposed by the British mathematician Hamilton in the 19th century.It is widely used in mathematics,physics,engineering,material science,medicine,etc.A Hamiltonian system can be simplified by its first integrals.Therefore,how to determine and give the first integral of the Hamiltonian system is one of the classic and hot research topics in the field of dynamic system.This dissertation is about the integrals of Hamiltonian system in polar coordinates.What we discussed is as follows:Firstly,for the Hamiltonian system under study,it consists of kinetic en-ergy and potential energy.So the algebraic relations between the first integrals of motion are analyzed.Thereout,by discussing the second-order and third-order first integrals of Hamiltonian system,the corresponding five completely integrable systems in the Liouville sense are acquired.Secondly,the superintegrability of the five completely integrable systems we obtained are analyzed respectively.By searching for the second integrals,the corresponding maximum superintegrals are established and the form of potential functions in the original Hamiltonian systems are determined and the second-order,third-order first integrals are built.In addition,the associated polynomial Poisson algebras with their algebraic dependent relations are exhibited.Finally,the integrability of Hamiltonian system with n = 2 is studied in detail.The algebraic relations between the first integrals of motion,the second-order and third-order integrals allowed,and the corresponding first integrals of the superintegrable systems,Poisson algebra with their algebraic dependence relations,are given in this case.