A gauge invariant formulation of time-dependent dynamical symmetry mappings and associated constants of motion for Lagrangian particle mechanics. I

Abstract

In this paper (part I of two parts), which is restricted to classical particle systems, a study is made of time-dependent symmetry mappings of Lagrange’s equations (a) Λi(L) =0, and the constants of motion associated with these mappings. All dynamical symmetry mappings we consider are based upon infinitesimal point transformations of the form (b) χi=xi+δxi [δxi≡ξi(x,t) δa] with associated changes in trajectory parameter t defined by (c) t̄=t+δt [δt≡ξ0(x,t) δa]. The condition (d) δΛi(L) =0 for a symmetry mapping may be represented in the equivalent form (e) Λi(N) =0, where (f) Nδa≡δL+Ld (δt)/dt. We consider two subcases of these symmetry mappings which are referred to as R1, R2 respectively. Associated with R1 mappings [which are satisfied by a large class of Lagrangians including all L=L (χ̇,x)] is a time-dependent constant of motion (g) C1≡ (∂N/∂χ̇i) χ̇i −N+(∂/∂t)[(∂L/∂χ̇i) ξi−Eξ0]+γ1(x,t), where γ1 is determined by R1. The R2 subcase is the familiar Noether symmetry condition and hence has associated with it the well-known Noether constant of motion which we refer to as C2. For symmetry mappings which satisfy both R1 and R2 it is shown that (h) C1=∂C2/∂t+γ1. The various forms of symmetry equations and constants of motion considered are shown to be invariant under the Lagrangian gauge transformation (i) L→L′=L+dψ (x,t)/dt.