Westudy Noether symmetries of a class of non-minimally coupled scalar field in a background spatially flat Friedmann-Robertson-Walker (FRW) spacetime.Weexplore the model symmetries and its conserved currents and charges. Especially, the scaling symmetry, its possible break down and outcomes of such a symmetry breaking are treated in details. A suitable potential of the non-minimally coupled scalar field is adopted which is necessary to get a symmetric Lagrangian of the system including gravity, scalar field and ordinary matter density.Weuse the obtained charge and the adopted potential in the equations of motions to see the role of the non-minimal coupling (NMC) on the cosmic expansion.Westudy evolution of the scalar field in the phase space of the model and explore the stability of the obtained critical point. In this manner we derive a relation that relates the cosmological constant and gravitational constant via a unique identity which reflects the scaling symmetry breaking in the space (a,j).
