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.