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

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) Λ_i(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) $\mathrm{t̄}$ =t+δt [δt≡ξ⁰(x,t) δa]. The condition (d) δΛ_i(L) =0 for a symmetry mapping may be represented in the equivalent form (e) Λ_i(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.