dorsal/arxiv
View SchemaMethod of Replacing the Variables for Generalized Symmetry of D'Alembert Equation
| Authors | G. A. Kotel'nikov |
|---|---|
| Categories | |
| ArXiv ID | physics/0012018 |
| URL | https://arxiv.org/abs/physics/0012018 |
| Journal | International Journal of Mathematics and Mathematical Sciences, 31 (2002) 149-155 |
Abstract
By symmetry of the partial differential equation L'\phi'(x')=0 with respect to the variables replacement x'=x'(x), \phi'=\phi'(\Phi\phi) it is advanced to understand the compatibility of engaging equations system A\phi'(\Phi\phi)=0, L\phi(x)=0, where A\phi'(\Phi\phi)=0 is obtained from the initial equation by replacing the variables, L'=L, \Phi(x) is some weight function. If the equation A\phi'(\Phi\phi)=0 may be transformed to the form L(\Psi\phi)=0, where \Psi(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 \Phi(x)=\phi'(x'\to x)/\phi(x).
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"abstract": "By symmetry of the partial differential equation L\u0027\\phi\u0027(x\u0027)=0 with respect\nto the variables replacement x\u0027=x\u0027(x), \\phi\u0027=\\phi\u0027(\\Phi\\phi) it is advanced to\nunderstand the compatibility of engaging equations system A\\phi\u0027(\\Phi\\phi)=0,\nL\\phi(x)=0, where A\\phi\u0027(\\Phi\\phi)=0 is obtained from the initial equation by\nreplacing the variables, L\u0027=L, \\Phi(x) is some weight function. If the equation\nA\\phi\u0027(\\Phi\\phi)=0 may be transformed to the form L(\\Psi\\phi)=0, where \\Psi(x)\nis the weight function, the symmetry will be named the standard Lie symmetry,\notherwise the generalized symmetry.\n It is shown that with the given understanding of the symmetry, D\u0027Alembert\nequation for one component field is invariant with respect to any arbitrary\nreversible coordinate transformations x\u0027=x\u0027(x). In particular, they contain the\ntransformations of the conformal and Galilei groups realizing the type of\nstandard and generalized symmetry for \\Phi(x)=\\phi\u0027(x\u0027\\to x)/\\phi(x).",
"arxiv_id": "physics/0012018",
"authors": [
"G. A. Kotel\u0027nikov"
],
"categories": [
"physics.class-ph",
"math-ph",
"math.MP"
],
"journal_ref": "International Journal of Mathematics and Mathematical Sciences, 31\n (2002) 149-155",
"title": "Method of Replacing the Variables for Generalized Symmetry of D\u0027Alembert Equation",
"url": "https://arxiv.org/abs/physics/0012018"
},
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