The Noether Bessel Hagen Symmetry Approach for Dynamical Systems

The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential 𝑉(𝜑). We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.