Geometric theory of non-regular separation of variables and the bi-Helmholtz equation
| Autor: | Raymond G. McLenaghan, Basel Jayyusi, Claudia Maria Chanu |
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| Rok vydání: | 2021 |
| Předmět: |
(pseudo-)Riemannian manifolds
Hamilton-Jacobi equation Laplace-Beltrami operator Variable separation Physics and Astronomy (miscellaneous) Helmholtz equation 010308 nuclear & particles physics 010102 general mathematics Multiplicative function Mathematical analysis Separation of variables FOS: Physical sciences Mathematical Physics (math-ph) 01 natural sciences Hamilton–Jacobi equation Geometric group theory Laplace–Beltrami operator 0103 physical sciences 0101 mathematics Mathematical Physics Mathematics |
| Zdroj: | International Journal of Geometric Methods in Modern Physics. 18 |
| ISSN: | 1793-6977 0219-8878 |
| Popis: | The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system in any pseudo-Riemannian space. The equation is studied in the four coordinate systems in the Euclidean plane where the Helmholtz equation and hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz equation admits non-trivial non-regular separation in both Cartesian and polar coordinates, while it possesses only trivial separability in parabolic and elliptic-hyperbolic coordinates. The results are applied to the study of small vibrations of a thin solid circular plate of uniform density which is governed by the bi-Helmholtz equation. 25 pages |
| Databáze: | OpenAIRE |
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