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Oscillatory motions at constant amplitude admit of an especially simple description of their stability in phase space: orbital stability is equivalent to stability of the amplitudes alone, regardless ...

Nonorthogonal R-separable coordinates for four-dimensional complex Riemannian spaces

J. Math. Phys. 22, 42 (1981); doi:10.1063/1.524753

Issue Date: January 1981

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E. G. Kalnins
Mathematics Department, University of Waikato, Hamilton, New Zealand

W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
We classify all R-separable coordinate systems for the equations Delta4Psi=[script J]<sup>4</sup><sub>i, j = 1</sub>g−1/2 [partial-derivative] j(g1/2gi j[partial-derivative]iPsi) =0 and [script J]<sup>4</sup><sub>i, j = 1</sub>gi j[partial-derivative]iW[partial-derivative] jW =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 R-separable system for the Laplace equation Delta4Psi=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Delta4Phi=lambdaPhi on one of the manifolds E4, S1×S3, S2×S2, and S4. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
Permalink: http://link.aip.org/link/?JMAPAQ/22/42/1
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KEYWORDS and PACS

Keywords
PACS
  • 02.40.Ky
    Mathematical methods in physics Geometry, differential geometry, and topology Riemannian geometries
  • 04.20.Cv
    Relativity and gravitation General relativity Fundamental problems and general formalism
  • YEAR: 1981

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (11)
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  1. E. G. Kalnins and W. Miller, Jr., Trans. Am. Math. Soc. 244, 241 (1978).
  2. H. P. Robertson, Math. Ann. 98, 749 (1927).
  3. L. P. Eisenhart, Ann. Math. 35, 284 (1934).
  4. C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., Commun. Math. Phys. 59, 285 (1978).
  5. E. G. Kalnins and W. Miller, Jr., Proc. R. Soc. Edinburgh Sect. A 79, 227 (1977).
  6. C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., Trans. Amer. Math. Soc. 242, 355 (1978).
  7. E. G. Kalnins and W. Miller, Jr., J. Diff. Geom. (to appear 1980).
  8. E. G. Kalnins and W. Miller, Jr., J. Phys. A 12, 1129 (1979).
  9. E. G. Kalnins and W. Miller, Jr., J. Math. Anal. Appl. (to appear).
  10. L. P. Eisenhart, Riemannian Geometry, 2nd printing (Princeton U.P., Princeton, 1949), pp. 89–92.
  11. W. Miller, Jr., Symmetry and Separation of Variables (Addison-Wesley, Reading, Mass., 1977).

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