The construction of conserved quantities for linearly coupled oscillators and study of symmetries about the conserved quantities

In this paper, the conserved quantities are constructed using two methods. The first method is by making an ansatz of the conserved quantity and then using the definition of Poisson bracket to obtain the coefficients in the ansatz. The main procedure for the second method is given as follows. Firstly, the coupled terms in Lagrangian are eliminated by changing the coordinate scales and rotating the coordinate axes, secondly, the conserved quantities are obtain in new coordinate directly, and at last, the conserved quantities are expressed in the original coordinates by using the inverse transform of the coordinates. The Noether symmetry and Lie symmetry of the infinitesimal transformations about the conserved quantities are also studied in this paper.