Exact Solutions for Cosmological Models with a Scalar Field

We consider the existence of a Noether symmetry in the scalar-tensor theory of gravity in flat Friedman Robertson Walker (FRW) cosmology. The forms of coupling function $\omega (\varphi )$ and generic potential $V(\varphi )$ are obtained by requiring the existence of a Noether symmetry for such theory. We derive exact cosmological solutions of the field equations from a point-like Lagrangian.