Notes on the symmetries of systems of differential equations

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Abstract
The concept of symmetry of the solutions of a system of differential equations is clarified. The functional character of the symmetry transformations is stressed in contrast with the pointlike character of the ordinary transformations considered by Lie. It is shown that any differential equation of arbitrary order possesses infinitely many symmetries, in strong contrast with a general theorem denying the existence of pointlike transformations of symmetry for an arbitrary differential equation of order greater than one. The relevance of local differential symmetries in theoretical mechanics is discussed, and some unsolved questions are raised.