The Noether Symmetry Approach: Foundation and applications. The case of scalar-tensor Gauss-Bonnet gravity

We sketch the main features of the Noether Symmetry Approach, a method to reduce and solve dynamics of physical systems by selecting Noether symmetries, which correspond to conserved quantities. Specifically, we take into account the vanishing Lie derivative condition for general canonical Lagrangians to select symmetries. Furthermore, we extend the prescription to the first prolongation of the Noether vector. It is possible to show that the latter application provides a general constraint on the infinitesimal generator $\xi$, related to the spacetime translations. This approach can be used for several applications. In the second part of the work, we consider a gravity theory including the coupling between a scalar field $\phi$ and the Gauss-Bonnet topological term $\mathcal{G}$. In particular, we study a gravitational action containing the function $F(\mathcal{G}, \phi)$ and select viable models by the existence of symmetries. Finally, we evaluate the selected models in a spatially-flat cosmological background and use symmetries to find exact solutions.