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

Abstract

The primary purpose of this paper is to show that infinitesimal velocity-dependent symmetry mappings [(a) x̄i =xi +δxi, δxi ≡ ξi(ẋ,x,t)δa with associated change in path parameter (b) t̄=t+δt, δt ≡ ξ0(ẋ,x,t)] of classical (including relativistic) particle systems (c) Ei(ẍ,ẋ,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(ẋ,x,t) +ẋiξ0,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 velocity-dependent Noether mapping theory is included in order to compare for Lagrangian systems the procedure for obtaining the characteristic form (e) of the general mappings with the procedure for obtaining the well-known formula (f) ZiN =Hij(ẋ,x,t)∂Z/∂ẋ j (Z ≡ constant of motion), characteristic of velocity-dependent Noether mappings. It is shown how any given velocity-dependent symmetry mapping function Zi(ẋ,x,t) (including Noether mappings) can be expressed in the form (e). A collection of variational formulas and identities is derived in order to develop from first principles the velocity-dependent symmetry mapping theory. Throughout, comparisons are made between velocity-dependent and velocity-independent symmetry theory.