Surprisingly, an interesting property of the Noether charge that it is by itself invariant under the
corresponding symmetry transformation is never discussed in quantum field theory or classical
mechanics textbooks we have checked. This property is also almost never mentioned in articles
devoted to Noether’s theorem. Nevertheless, to prove this property in the context of Lagrangian
formalism is not quite trivial and the proof, outlined in this article, can constitute an useful and
interesting exercise for students.