## Abstract

We classify all Râ€�separable coordinate systems for the equations Î”_{4}Î¨=J^{4} _{i,â€‰j=1}g^{âˆ’1/2} âˆ‚_{â€‰j}(g^{1/2}g^{iâ€‰j}âˆ‚_{i}Î¨) =0 and J^{4} _{i,â€‰j=1}g^{iâ€‰j}âˆ‚_{i}Wâˆ‚_{â€‰j}W =0 with special emphasis on nonorthogonal coordinates, and give a group theoretic interpretation of the results. For flat space we show that the two equations separate in exactly the same coordinate systems and present a detailed list of the possibilities. We demonstrate that every Râ€�separable system for the Laplace equation Î”_{4}Î¨=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Î”_{4}Î¦=Î»Î¦ on one of the manifolds E_{4}, S_{1}Ã—S_{3}, S_{2}Ã—S_{2}, and S_{4}.

Original language | English (US) |
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Pages (from-to) | 42-50 |

Number of pages | 9 |

Journal | Journal of Mathematical Physics |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1981 |

### Bibliographical note

Copyright:Copyright 2018 Elsevier B.V., All rights reserved.

## Keywords

- CONFORMAL MAPPING
- COORDINATES
- HAMILTONâJACOBI EQUATION
- LAPLACE EQUATION
- METRICS
- RIEMANN SPACE
- SYMMETRY