Noether symmetry approach and its generation of various conserved quantities of two dimensional dynamical system

In our current research, we employ Noether symmetry approach to generate conserved quantities for a two-dimensional Euler-Lagrange system depicting the dynamics of a simple harmonic oscillator and a simple harmonic oscillator with linear external driving force. By introducing a pair of Lagrangians we have found three novel types of invariant quantities such as Mei conserved quantity, Lie conserved quantity and Noether conserved quantity reminiscent to those previously reported by Fang et al. (2010) [1] (Phys. Lett. A, 374 (2010) 1806-1811) and further extended by Nucci (2011) [2] (Phys. Lett. A, 375 (2011) 1375-1377) for the uncoupled system under consideration. Generally, a system of linear second-order Euler-Lagrange equations admits a 15-dimensional algebra of Lie point symmetries amongst which maximum 8 could be Noether symmetries and consequently Noether's theorem assists in computing 8 associated first integrals. Here, however, we have achieved 9 distinct Noether symmetries while 11 distinct associated conserved quantities by introducing two Lagrangians formalism. In these 11 conserved quantities two (Lie conserved quantity and Noether conserved quantity) are induced by one Lagrangian and one (Mei conserved quantity) is induced by other Lagrangian. Interestingly, these three conserved quantities are reminiscent to those presented in [1] and remaining are similar to those found in [2]. We have also utilized the conservation laws to obtain the analytical solution of uncoupled system of oscillators. Furthermore, the current study adds further to what already available in the literature along with the physical explanation of the dynamical system under consideration.