Noether's theorem in classical mechanics revisited

Noether's [ 1] theorem, presented in 1918, is one of the most beautiful theorems in physics. It relates symmetries of a theory with its laws of conservation. Many modern textbooks on quantum field theory present a pedagogical version of the theorem where its power is demonstrated. The interested reader is referred to the detailed discussion due to Hill [ 2]. Despite the great generality of this theorem, few authors present its version for classical mechanics. See for example the work of Desloge and Karch [ 3] using an approach inspired in the work of Lovelock and Rund [ 4]. Several authors demonstrate Noether's theorem starting from the invariance of the Lagrangian [ 5, 6], but in this case it is not possible to obtain the energy conservation law in a natural way. In this paper, the theorem is proved imposing invariance of the action under infinitesimal transformation, opening the possibility of extending Noether's theorem in classical mechanics to include the energy conservation. In section 2, the Euler - Lagrange equation is rederived. In section 3, Noether's theorem is proved, in section 4 several applications are presented and in section 5 Noether's theorem is extended and energy conservation obtained.