Scalar field, nonminimal coupling, and cosmology

We study the dynamics of a flat Friedmann-Robertson-Walker universe filled with a self-interacting scalar field nonminimally coupled to the gravitational field. Dynamical equations for the system can be derived from a pointlike Lagrangian. For this system an additional Noether symmetry exists provided that the coupling constant xi is equal to 0 or 1/6. When xi = 1/6 the scalar potential has to be constant. In this case we obtain an exact solution. We also analyze the behavior of the scalar field when xi is not 0 or 1/6. Most of the considered solutions are unphysical but there exists a very interesting case in which the effective cosmological constant is rapidly changing, which might lead to inflation.