  We prove by symmetry properties that the Lagrangian of a free point-mass is a
quadratic function of the speed in the non-relativistic case, and that the
action of the free point-mass between two spacetime points is the proper time
passed in the relativistic case. These well known facts are proved in a
mathematically rigorous way with a frame independent treatment based on
spacetime models introduced by Matolcsi. The arguments show that these results
are not obvious at all, some common beliefs can be refuted by explicit
counterexamples. In our treatment the similarity of non-relativistic and
relativistic cases is apparent.
