Classification of the Horndeski cosmologies via Noether symmetries

Adopting Noether point symmetries, we classify and integrate dynamical systems coming from Horndeski cosmologies. The method is particularly effective both to select the form of Horndeski models and to derive exact cosmological solutions. Starting from the Lagrangians selected by the Noether symmetries, it is possible to derive several modified theories of gravity like f ( R ) gravity, Brans–Dicke gravity, string inspired gravity and so on. In any case, exact solutions are found out.