Method of Replacing the Variables for Generalized Symmetry of D'Alembert Equation

Abstract

By symmetry of the partial differential equation L'ϕ'(x')=0 with respect to the variables replacement x'=x'(x), ϕ'=ϕ'(Φϕ) it is advanced to understand the compatibility of engaging equations system Aϕ'(Φϕ)=0, Lϕ(x)=0, where Aϕ'(Φϕ)=0 is obtained from the initial equation by replacing the variables, L'=L, Φ(x) is some weight function. If the equation Aϕ'(Φϕ)=0 may be transformed to the form L(Ψϕ)=0, where Ψ(x) is the weight function, the symmetry will be named the standard Lie symmetry, otherwise the generalized symmetry. It is shown that with the given understanding of the symmetry, D'Alembert equation for one component field is invariant with respect to any arbitrary reversible coordinate transformations x'=x'(x). In particular, they contain the transformations of the conformal and Galilei groups realizing the type of standard and generalized symmetry for Φ(x)=ϕ'(x'\to x)/ϕ(x).