关键词: 两自由度微扰力学系统, 一阶近似守恒量, 泊松括号法, 一阶近似对称性

Abstract: A Poisson bracket method to obtain the first order approximate conserved quantities of two-dimensional perturbed mechanical system is proposed. We consider the perturbed Hamiltonian function as the combination of Hamiltonian function of unperturbed system and the perturbed term. First, according to the peculiarity of unperturbed system, we select a suitable method to obtain the exact conserved quantities of unperturbed system. Second, we calculate the first order perturbed terms of conserved quantities by using Poisson bracket and the character of partial differential equations. Finally, according to the characters of Noether symmetries, Lie symmetries and Mei symmetries, the first order approximate Noether symmetries, approximate Lie symmetries and approximate Mei symmetries of the first order approximate conserved quantities can be obtained. A perturbed two-dimensional isotropic harmonic oscillator is studied in this paper, and three first order approximate conserved quantities are obtained by using Poisson bracket method, and the first order approximate symmetries of three first order approximate conserved quantities are either approximate Noether symmetries or approximate Lie symmetries and Mei symmetries.

Key words: two-dimensional perturbed mechanical system, first order approximate conserved quantities, Poisson bracket method, first order approximate symmetries