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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 $V(\phi )$. 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.

Several applications of Lie symmetries and its generalisation are presented: from turning butterflies into tornados, to its applications in epidemics, population dynamics, and ultimately converting classical problems into the quantum realm. Applications of nonclassical symmetries are also illustrated.

Thanks to the introduction of a general Lagrangian, we derive a simple expression for Mercury perihelion advance (MPA) which can be applied to many historical examples, such as Poincaré’s 1906 application of his relativistic action, Einstein’s probable test of his 1912 metric, Einstein-Besso’s 1914 application of the Entwurf … as well as to modern presentations. We argue that Einstein’s failure to derive in his Zurich Notebook the right equations of general relativity has a physical rather than mathematical origin, which amounts to an incomplete application of the 1895 Lorentz Transformations to his 1911 formulation of the Equivalence Principle. MPA could have been already in his hands in 1912.