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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) 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) χ^{ i }=*x* ^{ i }+δ*x* ^{ i } [δ*x* ^{ i }≡ξ^{ i }(*x*,*t*) δ*a*] with associated changes in trajectory parameter *t* defined by (c) =*t*+δ*t* [δ*t*≡ξ^{0}(*x*,*t*) δ*a*]. 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) *N*δ*a*≡δ*L*+*L* *d* (δ*t*)/*d* *t*. We consider two subcases of these symmetry mappings which are referred to as *R* _{1}, *R* _{2} respectively. Associated with *R* _{1} mappings [which are satisfied by a large class of Lagrangians including all *L*=*L* (χ̇,*x*)] is a time‐dependent constant of motion (g) *C* _{1}≡ (∂*N*/∂χ̇^{ i }) χ̇^{ i } −*N*+(∂/∂*t*)[(∂*L*/∂χ̇^{ i }) ξ^{ i }−*E*ξ^{0}]+γ_{1}(*x*,*t*), where γ_{1} is determined by *R* _{1}. The *R* _{2} 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 *C* _{2}. For symmetry mappings which satisfy both *R* _{1} and *R* _{2} it is shown that (h) *C* _{1}=∂*C* _{2}/∂*t*+γ_{1}. 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*+*d*ψ (*x*,*t*)/*d* *t*.

© 1976 American Institute of Physics

Published online 28 August 2008

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2008-08-28

2016-10-28

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