Symmetry-Based Selection of Gravitational Lagrangians via Noether Approach

Starting from the Noether first theorem, we discuss a criterion for identifying physically consistent gravitational models. In particular, we demonstrate that applying the Lie derivative to a point-like Lagrangian makes it possible to determine the underlying symmetries, their associated generators, and the corresponding conserved quantities. This method is then generalized through its first prolongation and applied to several point-like Lagrangians, with special attention to f(R) gravity in a cosmological setting. In each example, the existence of symmetries results in a simplification of the dynamical system, enabling the integration of the equations of motion and the derivation of exact solutions. It is worth noticing that the existence of symmetries is always related to physically consistent models.