The Research On Integrability Of Several Hamiltonian Systems With Two Degrees Of Freedom

A Hamiltonian system is completely integrable in the sense of Liouville if it has sufficient first integrals satisfying corresponding conditions.The su-perintegrability of this Hamiltonian system can further studied on the basis of integrability.Therefore,the first integral is the key to the research of integra-bility and superintegrability of Hamiltonian system,and the construct of first integral is also crucial.At present,the research on the integrability of Hamil-tonian system is mostly based on constant(or zero)curvature space,however variable curvature space is more common,so the analysis of Hamiltonian systems with variable curvature is also very important.In this dissertation,four finite dimensional Hamiltonian systems are studied,in which the first two are with constant curvature,and the last two are with variable curvature.The outline of the dissertation is as followsFirstly,the Killing vectors corresponding to kinetic energy in the first sys-tem are determined,the quadratic integrals and potential function of system are constructed by using these vector fields,and three superintegrable systems in 2 dimensional constant curvature space are given.In addition,the Poisson alge-bra for each superintegrable system is exhibited and the polynomial algebraic dependence relations of the first integrals are given explicitlySecondly,the integrability and superintegrability of the second Hamiltonian system are analyzed,and the two cases of n=2 and n?2 in this system are discussed separately.To make the derivations be much easier,the original system is transformed from polar coordinate system to cartesian coordinate system by coordinate transformation.In the Cartesian coordinate system,the conformal Killing vector in the 2 dimensional space is used to construct the quadratic integrals and potential function of the system,and then five superintegrable Hamiltonian systems corresponding to the 2 dimensional constant space and their corresponding Poisson algebra are obtained.Last,all the superintegrable systems obtained here are converted to the original polar coordinate systemFinally,the quadratic integrals and potential function are constructed sep-arately by using conformal Killing vectors in 2 dimensional space to determine the integrability of the third and the forth systems.New quadratic integrals are further introduced in the two systems,and then three superintegrable systems and Poisson algebra of the two systems in 2 dimensional variable curvature space were established respectively.