  Here we consider scale invariant dynamical systems within a classical
particle description of Lagrangian mechanics. We begin by showing the condition
under which a spatial and temporal scale transformation of such a system can
lead to a symmetry by leaving the action unchanged. We then derive the form of
all analytic Lagrangians that possess such a symmetry under change of scale.
Finally, we write down the Noether charges of all analytic Lagrangians that are
associated with a scale invariant action.
