On Lie point symmetries in mechanics

We present some remarks on the existence and the properties of Lie point symmetries of nite dimensional dynamical systems expressed either in Newton-Lagrange or in Hamilton form. We show that the only Lie symmetries admitted by Newton-Lagrange-type problems are essentially linear symmetries, and construct the most general problem admitting such a symmetry. In the case of Hamilton problems, we discuss the diierences and the relationships between the existence of time-independent Lie point symmetries and the invariance properties of the Hamiltonian under coordinate transformations.