Research In Integrability Of Several Classes Of Autonomous Systems Based On Lie Group Theory

Based on Lie group theory, the integrability of second order, third order and n-th order autonomous system is studied. The main contents and results of this dissertation are included as following aspects:(1) The general form of generators of one-parameter Lie groups accepted by second order exact automatic system is given. Then, some properties of the system are discussed. At the same time, the general form of generators of one-parameter Lie group which is accepted by two particular classes of second order automatic systems is given, thus the integrating factor of the system is constructed.(2) Base on Lie group theory, the method to obtain the integrating factors of the third order autonomous system by using of two one-parameter Lie groups accepted by the system is given and shown.(3) For the n-th order system, the relation between integrating factors and first integrals of the system is discussed when it is conservative system; At the same time, the sufficient and necessary condition for the existence of integrating factors of the system is given and proved when it is non-conservative system.