一维减幅-增幅谐振子的守恒量与对称性

从一维减幅-增幅谐振子的运动微分方程出发得到系统的运动积分常数，从而得到系统的Lagrange函数和Hamilton函数,再根据Hamilton函数的形式假定守恒量的形式，由Poisson括号的性质得到了系统的三个守恒量，并讨论与三个守恒量相应的无限小变换的Noether对称性与Lie对称性.还对守恒量与对称性的物理意义作了合理的解释.关键词：一维减幅-增幅谐振子守恒量Noether对称性Lie对称性

The study of symmetries and conserved quantities for one-dimensional damped-amplified harmonic oscillators

In this paper, a constant of motion of one-dimensional damped-amplified harmonic oscillators is derived from Newton's equations, and the Lagrangian and the Hamiltonian of system are expressed in term of the constant of motion. According to the expression of the Hamiltonian, we make an ansatz for the conserved quantity and then three conserved quantities are obtained by using the definition of Poisson bracket. The Noether symmetry and Lie symmetry of the infinitesimal transformations of the three conserved quantities are studied and the essence of symmetries and conserved quantities are also explained in this paper.