Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. I. Velocity‐dependent mappings of second‐order differential equations

The primary purpose of this paper is to show that infinitesimal velocity‐dependent symmetry mappings [(a) xi =xi +δxi, δxi  ≡ ξi(x,x,t)δa with associated change in path parameter (b) t=t+δt, δt ≡ ξ0(x,x,t)] of classical (including relativistic) particle systems (c) Ei(x,x,x,t) =0 are expressible in a form with a characteristic functional structure which is the same for all dynamical systems (c) and is manifestly dependent upon constants of motion of the system. In this characteristic form the symmetry mappings are determined by (d) ξi =Zi(x,x,t) +xiξ0,ξ0 arbitrary; the functions Zi appearing in (d) have the form (e) Zi =BAgiA(C1,...,Cr; t), 0≤r≤2n, A=1,...,2n, where the BA are arbitrary constants of motion and the C’s appearing in the functions giA are specified constants of motion.A procedure is given to determine the giA. For Lagrangian systems it follows that velocity‐dependent Noether mappings are a subclass of the above‐mentioned general symmetry mappings of the form (a)–(e). An analysis of v...