A study of phantom scalar field cosmology using Lie and Noether symmetries

The paper deals with phantom scalar field cosmology in Einstein gravity. At first using Lie symmetry, the coupling function to the kinetic term and the potential function of the scalar field and the equation of state parameter of the matter field are determined and a simple solution is obtained. Subsequently, Noether symmetry is imposed on the Lagrangian of the system. The symmetry vector is obtained and the potential takes a very general form from which potential using Lie Symmetry can be obtained as a particular case. Then we choose a point transformation $(a,\phi)\rightarrow(u,v)$ such that one of the transformed variables (say u) is a cyclic for the Lagrangian. Using conserved charge (corresponding to the cyclic coordinate) and the constant of motion, solutions are obtained.